Welcome back again to the second lecture in this module of linear first and second order equations. In the first lecture we have completely studied the first order linear equations and we essentially seen that the first order linear equations both homogeneous and non-homogeneous can be essentially converted into the integral calculus problem. This is actually general feature of that another class of problem called exact differential equations. So, basically we have seen that first order linear equations can be put it in the exact differential form if necessary by multiplying by a suitable function called integrating factor. So, there is a complete study of the first order linear equations, but such an easy way of determining the solutions to second order linear equations is not available, but before going to the second order linear equations we will introduce in this lecture the concept of exact differential equations. And there is another interesting class which probably most of you are familiar called separable equation. The separable equations also can be integrated out straight way and it falls in the category of exact differential equation. So, I will give you one or two two examples of separable equations and then we will move on to the exact differential equations in this lecture. So, separable equations, separable equations. So, a differential equation of the form if you have if you can write a differential equation of the form d y by d t is equal to h of t into g of i. So, this equation need not have to be linear at all this equation need not have to be linear, but it is in the separable form h t the independent variable and independent variables are separated out. So, you can write formally this one in the form d y by d y d y by g of y is equal to d t of h of t. And then one can integrate because it is an integral one can actually integrate and then find the solutions okay. So, you will this implies you can have integral of d y by d y g of y is equal to integral of d t by h t plus a constant which can be solved. So, this is also an easy class of equations from the non-linear equations in general which can be solved easily. So, we will start with a familiar example which all of you are already familiar. So, for example you want to have an example simple example are the complicated examples you can do it, but the idea is not all the time to make the ideas clear rather than the getting into the complications of the integrals here. So, look at the easier equation t of y which you already seen it this can be solved immediately by d y by d y this in the separable form here h t is equal to t g of y is equal to y. So, h d y by d y by y is equal to d t by d y by d y is equal to t d t actually not t d y that implies you have to always do this procedure of log mode y in general log mode y is equal to t square by 2 plus a constant. And then you do the normal procedure mode y is equal to it is a constant say c 1 you can put it you can write this mode y is equal to some constant c into e power d square by 2, but then the same procedure which I have told this can be brought it here and is a continuous function a continuous function whose modulus is constant implies finally y is of the form c e power t by square by 2 this all of you are familiar and it is much more easy. So, I will go to the next example another example interesting. So, we will have 1 or 2 examples in this situation example when you go to it is a slightly different equation of the form d y by d t. So, there is a change d y by d t equal to minus t by y of course, this is not in the category of the normal equations because this function as a function of t and y is not continuous at here. So, this is a kind of this is in the category of singular differential equation. So, if you write this equation this is of the form y d y by d t is equal to minus t or in the formal notation of symbolic integral is nothing, but to minus t d t. So, naturally you can expect trouble at this at the origin because as I told you the moment there are coefficients whose vanishes it comes what are called the singular equation in general we do not treat with the singular equations, but in other modules you will see some special class of singular equations if our time permits. So, if you go here. So, naturally you can expect trouble near the origin which is exhibited that is why we want to show this example. So, if you integrate that one you will get y square plus some constant plus t square equal to constant you see that is a general solution for this first order equation, but we want to understand this graph these are nothing, but circles you see this equation of the circle in the implicit form. So, I want to understand the solution curves here. So, if you plot if you are trying to solve this equation it is not possible to solve uniquely. So, you will have different solutions and you have to understand. So, you have y t if you write it this will be square root of plus or minus c minus t square you see and this if you want to have a real solution then if t has to be that means t square has to be less than c or less than v equal to c. So, you see 0 less than t square less than v equal to c. So, that shows that the solution exists you have the solution exists solution for mod t less than v equal to c. So, if you are trying to plot these curves. So, you have to branch out. So, if you have a this will be more clear. So, if you have a point here. So, this solution curves are something like circles these are circles actually. So, you have solution curves like this similarly you will have solution curves from here. So, these are two different solution curves for the initial value problem. So, let me put it that way. So, you will have solution curves coming that depends on each c. So, you will have depending on you will have a minus root c here we will find out what is that minus root c and you will have roots. So, let us look at an initial value problem. So, if you want if. So, if you have the initial value problem I v p and suppose you have described y at t naught is equal to y naught. So, if that is a to determine your c here your this will imply your c will satisfy y naught square plus t naught square. So, this will imply your solution is y square a plus t square is equal to y naught square plus t naught square for your information. So, if you try to see this thing. So, if so it depends on where y naught is depending on the sign of y naught which branch of the solution which is the solution defined thing. So, suppose assume that. So, let me plot the curves in the same page. So, that we will complete this particular problem. So, suppose your y naught is positive what will happen. So, if you have this if you try to plot this curve again come back to the let me take another color. So, if you plot this. So, you have your t naught here somewhere your t naught it can be t naught can go negative also and your y naught here. So, if you have y naught. So, you will have the circle. So, if your y naught is here. So, this is your y naught. So, your t naught y naught. So, you see the solution x is only this is nothing but square root of y naught square plus t naught square this will be this point will be minus of square root of y naught square plus t naught square. So, you see. So, you have your solutions from existing in the interval mod t the solution x is mod t less than or equal to plus square root of y naught square plus t naught square is an interesting example. On the other hand if you if this is the case y naught is negative positive if you have y naught is negative you will have the other branch you will see you will have the other branch where you are here this is here y naught here y naught. So, that is a solutions will come like that. So, the solution is defined from this interval. So, this of course, you cannot have anything you have trouble at the origin because of this singularities. So, you will have the solution. So, it is an interesting example of a separable equation one can deal with it. You can also use this equations to solve sometimes separable equations can be used to solve a second order equation. So, let me a present you one more example without details. Yeah, but second order equation. Suppose you have a simple equation for the form y square equal to we have not studied second order equation, but you do not need to know anything about the second order equation. Suppose you have this equation what you do is that you put v is equal to y prime then this equation will become v prime is equal to v prime equal to you have your equation t square v. So, you see you can this is a first order equation first order and you can solve for v t. So, you can write down your v t by the usual way some constant c 1 into e power t cube by 3. So, you can have your solution and now what is v t this is nothing but d y by d t. So, you have another first order equation which you can integrate and eventually you can write your solution y t is equal to c 1 integral of e power t cube by 3 plus some c t that is it another and you will get. So, such equations are can be solved. So, I will give a probably you can try another exercise for you exercise try the equation a simple equation y double prime y prime equal to t into 1 plus t find the solutions of the initial value problem y at 0 is equal to 1 y prime 0 is equal to 0 to 2. So, you can get a find the solution. So, this is because of this special form as you see that because of this special form there is no y here you are able to bring it to the same thing you can do it you can bring it down this equation you can bring it out the first order equation solve it and then you will get another first order equation and then find the solutions to it that is it with this 2 x 2 3 examples. Now, we will go to what is called the exact differential equations. So, in this lecture we will try to explain certain things about the exact differential equations and then we will move on to the second order linear equations and how to study second order linear equations. So, we will go to what is called exact differential equations. We are covering these things even though these are all important and these are the things taught in the university system. So, we want to go to that, but we are advice the students those who are taking this course to work out few more examples then I am explaining here to get familiarize these things. So, what we have seen? So, basically we have studied 3 types of equation. One is the integral calculus problem. Suppose we are given an equation of this form this is the integral calculus problem that is what I terminology I am using here integral calculus that is the first thing we have studied the second thing is what we have studied is the first order linear equation y prime plus P t y is equal to Q t this is what we have so far completed this is the first order linear equation and essentially what we have seen is that by multiplying suitably if necessary you can bring back to everything you can bring back to the integral calculus problem and third one what we have studied now is y prime is of the form this is non-linear, but separable form h of t into g of y separable form all these equations are in the category of this exact differential equations that is what we will be. So, we will want to formally study this exact as I said the most general form of your differential equation is f t y y prime equal to 0 you can define the terminology of exact differential equations even for this equation, but we are since we are not planning to deal with this more general equation our equation what we are considering is the standard one d y d t is equal to f of t y. So, let me start with the thing suppose there is a 2 variable function suppose we have there x is a function phi which is a function of 2 variable where you this is very important thing it is not that when I am defining phi v is restricted to and we do not treat that y is a depend independent variable of t y is not just y t right now phi is defined in a domain of the plane t y. So, where you treat both t and y are independent variable. So, it is a 2 variable function when I write y t then you are basically restricting your y t to the solution comes at present it is a function of 2 variables in the t y plane basically such that you can write this equation. So, let me define even for this thing or this full equation you can write it such that you can write your f of t y of t y prime of t you see now you are restricting as a total derivative that is what if you look at the studies of these 3 equations 1 2 3 essentially we have done this thing of d by d t of phi of t y t suppose there x is such a function satisfying this thing then that implies d by d t of phi of t y t equal to 0 in other words this can be integrated now so that is why phi t. So, the solutions are given in implicit form is constant. These are the solutions in implicit form. So, such differential equations are called such differential equations or called exact differential equations. So, we want exact differential equations. So, we want exact differential equations. So, we want conditions under which we want some verifying conditions. So, that we want to know whether a given differential equation is exact or not. If the given differential equation is exact you can immediately find the solutions and what we have seen is in the case of 1 2 3 whether in the calculus first order or separable equations you can do this quickly. And if not if the differential equation is not exact can you find a some multiplying factor can you multiply the differential equation. So, that the multiply differential equation is exact. So, we want to show you with that in mind that multiplication factor is also in mind we are going to see something more general. So, from we consider slightly more general equation than the previous one. So, we are going to consider something more. More means you can also the multiplying factors are taking in the account. So, in this differential equation we will see soon that we consider equations of the form equations of the form m t y plus n t y into d y by d t is equal to 0. You see where m and n are functions of 2 variable a functions of that is what I get functions of 2 variables t and y. So, it is depend on some domain in t y play t y we will define that of course, this equation. So, let me call this equation 1 which we are going to use it and also mark it here. So, you can immediately this equation equation 1 is 1 more general than more general than d y by d t is equal to f of t y let me call this 2. Why it is more general you can always write your equation 2 equation 2 in the form of 1 we can write 2 in the form 1 by choosing m equal to minus f n equal to 1 or f is equal to minus f is of the form m by n you see. But the advantage of choosing the equation 1 is that you will be able to even if you multiply by this is 1 choice of m and n. But what we can do is that you can also multiply by a multiplying factor the same multiplying factor for m and n and you get back the same f. So, the advantage is that you can write here in the form m in different ways with the different factors. And what we will see that for certain representations the equations may not be exact, but if you choose a appropriate multiplying factor is possible that you get the exactness of the differential equation. This is exactly we will be doing it. So, you can see that in this example. So, this gives you an equation 1 gives you an added advantage of representing this equation. In books I also want to make sure the students who are studying the unicycle system quite often this equations will be written in this form m d t plus n d y equal to 0. So, but I want to retain in this form. So, that we have do not have any ambiguity of d t d y. But basically this is that here. So, what we are going to say for example, let me give you one example here and then we will go to the theory a Q K example a simple example.  So, it is a, but this can be easily seen is not a very easy to see that this is nothing, but you can see that you do not need a much knowledge about it. This is nothing, but t if you differentiate you get one sign you differentiate this one then you will have the other factor d plus sin of t plus y. So, if you differentiate you get this one is differential product form and you will have that quickly you have to differentiate this one also that is what we will give you. If you differentiate this will have this two terms. So, this equation will reduce to this form and you can find that this gives you the solution in implicit form you see solution t plus sin of t plus y equal to constant. So, you can many of these things not that every differential equation is exact. So, we will go to. So, our interest is to find this equations. So, we want to when a differential equation exact when a differential equation one keep that in mind one for us one is this one and this is our differential equation two. So, we want to know when is the difference when a differential equation of the form one exact that is our form one exact that is all good. Suppose it is exact let us look for that is what you are look suppose we want to derive the thing suppose it is exact suppose one is exact what does that mean there x is phi assume the differentiability of mu such that m plus I am d y by d t equal to 0 d y by d t I am suppressing the variables t and y. So, if you want you can write m t y in the beginning. So, that gives you if this is exact if there x is a function this is equal to d by d t of phi t y. Now of course, y is all treated as a function of y. So, if you do the total derivative if you calculate this will be d phi by d t plus you will have d phi by d y into d y by d t this immediately you can this because of this one of course, from here you cannot conclude m is equal to d phi by d t n equal to d y d, but that gives you a necessary sufficient condition. So, you will have immediately a sufficient condition sufficient condition what is the sufficient condition if there x is phi smooth whatever smoothness you required such that m is equal to d phi by d t. So, you are looking for a two variable function. So, you are looking for a two variable function c phi such that a single two variable function phi such that m is its derivative with respect to the first argument d phi by d t and n is the derivative with respect to the second argument not d phi by d t d phi by d y that will implies then one is exact. So, you have. So, you want to know more about it then one is now. So, the question is that. So, you can immediately see that this thing. So, the so the question we want to so we want to post a question now in this question one given two functions m and n given two functions two functions in two variable m and n m is a function of two variable in a domain in t y plane n equal to n t y does there x is the question is that does there x is does there x is f function phi such that m is equal to d phi by d t. So, the question we formulated. So, at least we get it n is equal to d phi by d y. So, if you can do that you can answer this question you can answer this question. So, you have the you have the so you see. So, the question of controller a question of x hat the differentiability x hat differential equation can be answered if you can find that one. So, that is a question we want to answer. So, let me put it in a in the form of a theorem a simple theorem it is not a difficult theorem. So, I want to write a short. Suppose m n belongs to this notation probably you may study this is c 1 of d is nothing but continuously differentiable functions a function is continuous in a domain d where let us the d is a rectangle you can write it in r t y plane a b cross c d. So, you see it is a two variable function defined on a domain in r 2 r 2 is the t y plane and m n then you need smoothness that is is continuously differentiable then then there x is phi such that m is equal to d phi by d t n is equal to d phi by d y. So, this is a simple theorem this is an if and only if if and only if d phi by d y is equal to d phi by sorry you want interesting thing you want conditions in terms of the m n if and only if d m by d y is equal to d n by d t. So, you see the interesting fact here is the condition in terms of the given data m and n are functions given to you in the differential equation. So, you can verify a given differential equation the moment you put your differential equation is of the form n plus n d y by d t you can check that condition at easy condition to be checked adjusted differentiation which anybody can do it easily to verify that it is a exact differential equation or not. And the condition is not very sufficient because one way is trivial because if d m by d y is equal to d n by d t this is equal to saying that d square phi by d y d t equal to d square phi by d t d y in exchange of differentiation is true. So, you immediately get one part of it and then proving the other part is also not difficult.  So, assume one condition. So, we will have a we will have the thing one way proving it there x is phi that is why. So, we will prove this part first simply. So, assume phi x is assume phi x is satisfying m equal to d phi by d t and n equal to d phi by d y. Then that implies this is a trivial part I am telling d m by d y is equal to d square phi by d square d y d t by interchanging for smooth functions can be interchanged by interchanging you get this is equal to d n by d t. So, that is fine. So, that is so converse converse is the one you have to prove. So, you want to prove this one. So, assume d m by d y equal to d n by d t. Now, what we have to prove? We have to prove the existence of phi satisfying these two conditions. We want to find the existence of phi satisfying these conditions. So, if you look at here. So, if you look at this equation you want to satisfy these two conditions you want to do that one. So, what do we do? So, we have to achieve two constraints we have to find one v satisfying two equations m equal to d phi by d t and we also have to satisfy the same phi satisfying this thing. So, what do we do is that? So, whenever you have difficulty two objectives to be achieved first we concentrate first we keep aside one of the constraints and look at only the other constraints. So, we can see that whether we can find this thing. So, that is what we are doing it. So, look at the constraints one. So, take consider this equation you want to see that we do not know we want to know this. Thus there exists m t y is equal to d phi by d t d y you want to prove this one. So, look at here this equation that is the first idea of understanding this equation. When you look at this equation it is nothing but a first order rho d e in t and y appears like a parameter that is what it is a parameter that is what you have to see that. So, there is no differentiation with respect to y. So, you can see that this is nothing but a first order equation we use this d by d t partial d by d t to represent that there is another variable. So, this is an equation in t variable first order equation in t variable treating it as m is given to you and achieve. So, you have to just integrate. So, this is an integral calculus problem what we have studied in the last class is an integral calculus problem for the differential equation in t. So, if you integrate the. So, choose the burst wave that you choose phi such that phi t y is just integrate m that is a natural way to do integration f t y and integration is with respect to t. But of course, when you integrate there will be an integral constant that is where the difference comes when you have an integration constant and that constant will depending on this parameter. So, each fixed parameter y for each fixed parameter y you will have a constant h then the parameter changes this constant change. So, here you have a function of y and this constant is the unknown now. So, the constant unknown function in y. So, if you differentiate naturally if you differentiate d phi by d t you will get this differentiation you will bring it into d m by d t and this will vanish and your first equation is satisfactory. So, that is what you do it, but now you have the constraint of the second equation. So, now substitute this in the second equation. So, you differentiate it. So, you want to determine still you want to determine you want to determine h. So, to determine h you use the second condition of 1. So, differentiate. So, you have to differentiate now differentiate with respect to y that you will imply. So, to do a proper differentiation you have to differentiate with respect to y. Now, this is an integration with respect to the this is an integration with respect to t. So, the integral will not get cancelled. So, the integral will not get cancelled. So, you will have the integration coming here integral of d m by d y suppressing the argument and but d t. So, you have d t here and then you have to differentiate h you have to differentiate h with respect to y. So, that it is denoted by h prime of y because there is only one argument or you can write it d h by d y. And you want this to be we want n equal to d phi by d y. So, that implies. So, you have an equation. So, you have that implies n is equal to integral of m d m by d y d t h prime of y you see. So, that implies a differential equation in y variable. Now, that implies this you will get that is. So, this comes that is h prime of y is equal to n minus integral of d m by d y d t. So, you integrate. So, n m m i find you will be able to find this one. So, integrate integrate again it is an integral calculus problem. Integrate but integrate with respect to y you will get h of y is equal to integral of you have to do that integration n minus integral of d m by d y integral of d m by d y integrated with respect to t you see and then this integration is with respect to y. So, you have that one. So, what is your phi t y that implies your phi t y look at that. So, y t y so you have the your phi t y is here where is your phi t y is here you have an integral m t y h here. So, you will have. So, you will have your phi t y is equal to integral of m d t plus integral of n this is n d y and then minus integral of integral of d n by d t d y d t. So, we have started with the equation. So, we have started with the equation. So, first for m and then we used for m you can also do with the other way you can start with n first and then you can start with m and you will have a different form that does not matter that I will leave it as an exercise. So, you see instead of m first starting you start with n and then proceed. So, you will have h of y you will have some h of t and you proceed that way and you can write down this all this can be integrated out. So, you will have an phi and you will have the phi t y equal to constant or the solutions to this differential equation. So, with this definition because there is a necessary and sufficient condition I can redefine my definition. Now, at definition I can do the redefine at differentially. So, because I would prefer to give conditions in terms of the given data. The given data is m and n. So, I can define the exact differentiation the earlier definition if there exist a phi satisfying the conditions like m equal to d phi by d t and n equal to d phi by d y, but then a phi is coming into definition in the in the definition. So, but we have a definition of the exact differential equation the differential equation of the form the differential equation of the form m plus n d y by d t is said to be exact if d m by d y is equal to d m by d t. So, it is just an easy verification. So, if you want to determine a differential equation is exact or not you have to just to compute this differential equation. So, we will say for example, if you want to have an example again an example like you can have plenty of example, but let us say you want to have me is an artificially cooked up example, but any equation you can do it 3 y plus e power t. You have to put the differential equation if a general problem is given you have to put it in suitable way cos of y into d y by d t. This is a cooked up example. So, that is not a problem. So, what is m here 3 y plus e power t n equal to d y by d t. So, what is equal to 2 t plus cos y. So, you can compute your. So, what is your d m by d y is to compute is nothing, but 3 and what is your d n by d t is this would not be exact. So, you want to have a 3 t d n by d t. So, this is 2. So, you have 3 t. So, you have 3 d n by d t is also equal to 3. You see the differential equation is exact. So, how do you proceed to find phi you do that one a step by step you do it. Therefore, there exists phi such that you do not have to remember the formula. You can proceed it to solve the problem there exists phi such that m is equal to d phi by d t and n is equal to d phi by d y. You can integrate it out one by one. So, if you have what is d phi by d t you have to find out. So, you want your d phi by d t is equal to solve that d phi by d t is equal to m m is equal to 3 y plus e power t. You see which is a if you treat y as a variable this is a y as a parameter and t as a variable which will give you you just integrate it out out and you can write down 3 t y is equal to 3 y t I have done the computation, but I do not want to spend time here in computing, but you can compute it.  If I do that computation this will be 3 t plus h prime of y you see a q. So, that you will imply and that has to be you want this to be d phi by d y to be n that is how you want that to be n. So, you will have h prime of y is equal to n n is given to you already here and you do proceed that one you have a 3 t 3 t will get cancelled. So, it is nothing but sin you see you got that. So, that you will imply h of y is of the form cos y you see n just integrate m or minus cos y. So, you will have your phi t y is equal to 3 y t plus e power t plus sin y your cos y h of y is equal h prime of y is cos y right h is sorry h prime of y is equal to cos y here. So, h of y is equal to sin y here you can work with other things also that is not a problem you see. So, this equal to constant you will give you the solution. So, you can you can work out plenty of things like that. So, you can do that. So, the easier part is that you have a very easy condition in terms of d m by d y is equal to d n by d t that is the advantage the verifying condition to be that. But, if the differential equation is not exact that is going to be difficult and that is what we quickly in the next 10 minutes we are going to discuss with you a quick discussion and then that will be the we will complete this example. So, that again leads to what are called integrating factors integrating factors in general differential equation. So, the question is again question can we find a factor multi-factor equation multiplying a multiplying factor multiplying factor which you already seen in the linear first order differential equation. So, multiplying mu is a function of no t and y such that mu t y mu into m plus mu into n into d y by d t is equal to 0 is exact. So, this you are interested in finding is exact you see of course, this is if m plus n d y by d t equal is exact you do not have to do that. The question is coming when m plus n d y by d t is not exact. So, that means you do not have. So, this is the situation this is the situation situation when d m by d y not equal to d m by d t you see if d m by d is equal to if d m by d y is equal to d n by d t then that equation is going to be exact and you do not have to proceed further. So, now let us suppose this is exact suppose this equation let me call it 3 suppose 3 is exact what does that mean again suppose 3 is exact that is you have to satisfy this condition when m is replaced by mu m and n is replaced by mu n that is d by d y of mu m is equal to d by d t of mu n that is d by d y of mu that is d mu by d y m plus mu d n by d y is equal to d mu by d t n plus mu d n by d t. Of course, this is like a part if you want to determine this is like a partial differential equation and in general finding a solution to this equation is difficult may be by trial and error you may be able to compare thing. So, in general determining a solution to this differential equation is difficult and it is very rare. So, that is where the most general case ends for a very general situation you may not be able to find in general a solution to this differential by some trial and error if you are successful in finding the solution to that differential equation then you can do that one. But then there is a very interesting special case is still it may work special case the special case is actually interesting in the sense that sometimes when you demand little you may not get anything that is what the general thing, but on the other hand if you demand little more you may get something better and that is what is happening here. So, we are going to demand something more. So, we are going to demand mu is equal to mu of t is a function of t alone of course, you can demand mu equal to mu of y is a function of y alone and a similar analysis can be done function of t alone. So, if that is the case look at this equation if mu is a function of t alone then this term vanishes and then you will have an equation of first order for t that is why the advantage. So, we are removing by demanding either mu is a function of t alone or mu is a function of y alone we are removing the equation and it equation becomes a first order equation for ODA and which we can solve it that is what we will do that one. In that case what we will get is that you will get an equation of the form mu t into that will imply if you write that equation mu t into d m by d y minus d n by d t is equal we are not sure even this is solvable and a condition will be given soon. So, this will be now we can write mu prime because mu is a function of y alone. So, that implies mu prime of t by mu t is equal to d m by d y minus d n by d t into 1 over n you see. Can this equation be solved for mu, but then look at the interesting fact this portion is a function of t this is no alone function of t alone, but this is a function of both t and y because m and n are functions of t alone. So, if this happen to be that thing with the right hand side of this equation the right hand side of this equation is a function of t alone then this is a first order equation and it can integrate to do. So, assume if rather than assume if 1 over n let me write m of y for the simplicity minus of n of t is a function of function of t alone that is possible even though m and n are functions of both t and y the combined factor 1 over n m y minus n t can be a function of t alone then and you see and integrating factor of the form mu equal to mu n t can be determined you see you can do that thing. So, you see in that case so this can be done even with when m mu is a function of y l thing you will get something else here the mu will come these things the roles of d m by d y and d n by d t will be interchange 1 by n will become 1 by m if that happens to be. So, you have to verify that and this is valid because if equation is exact this going to be 0 d m by d y minus d n this is the situation and d m by d y minus d m by d t not equal to 0 and in that case you need to calculate that one and determine by 1 by n or 1 by m and see that which one is independent it is only function of t alone in the other case you see that it is a function of y alone in that case either mu of t can be determined or mu of y can be determined and you can multiply your differential equation with that mu t or mu of y to get the solution.  So, this equation can be solved I want you to solve this equation you can see that this equation is exact the second exercise is what you already seen I consider this linear first order equation consider d y by d t plus p t y is equal to q t derive the integrating factor which you will see using this derive the integrating factor you have seen the integrating factor already in the first order equation study earlier and the third one which I am going to stop it you see that the separable equations are exact separable equations are exact. So, with this we are going to finish this class. So, you have seen that the first order equation can be it will be a exact differential equation by finding a suitable integrating factor which you are already seen and the separable equations are always exact and in the next lecture we will get into the second order linear differential equation. Thank you.


I think the first thought when looking into a new philosophy is what can it do for me and is it worth my time to investigate? Well in this video let's talk about how you could benefit from the philosophy of Stoicism. As my personal favorite philosophy, Stoicism is in my opinion invaluable in the increasingly chaotic world that we live in. What about the Stoic philosophy makes this statement true? Well I think it's right to start with perhaps the most important lesson, which in turn will allow us to master the other teachings with these. The first lesson is that it will teach you how to find inner peace. Whether it be through the eloquent writings of Seneca or the refreshing candor of the meditations, you will find plenty of pieces to the puzzle of inner peace, an ancient map to equanimity laid out by our betters. These thoughts are there for the taking, unguarded and in abundance for all to share equally should they only rise to the occasion. With inner peace now within your power, Stoicism will then teach you how not to waste time by making the most of each day. It's no secret that procrastination has been or still is an issue in all of our lives, but how many of us consider that that hour that we just wasted could be the last of its kind? What man can you show me who places any value on his time, who reckons the worth of each day, who understands that he is dying daily? For we are mistaken when we look forward to death. The major portion of death has already passed. Whatever years be behind us are in death's hands. This quote by Seneca drives home the somber fact that we would be hard pressed to find more than a handful of individuals that use its precious resource to its utmost limitations. Stoicism will allow you the opportunity to take your place among these select few by showing you the worth of each second, minute, hour, day, week, month, and year. If you ask those who have been afforded the luxury of living their lives to old age, most will tell you that the only thing they wish they had more of is time. Next on the list of how you could benefit from Stoicism is that it will free you from the opinions of others. There is a reason that public speaking is a fear likely only third to loneliness and death, because as much as the mass of humanity would like us to believe that they could care less about the opinions of others, in reality most of them are likely terrified by the thought of ignominy or public rejection. Stoicism will teach you how to overcome this by illuminating the simple but seemingly surreptitious fact that worrying about what you cannot control is pointless. As long as you have done what is according to justice, that which is right and virtuous, and what your peers think is irrelevant and even if it wasn't, it lies outside of your control.  Copernicus, Bruno, Socrates, the list goes on.  So much value can be found in the literature written by or about the Stoics. We've already covered a fair share of invaluable lessons, but I would like to share with you one more crucial lesson from the Stoics, and that lesson is how to conquer death. One must only read Seneca's On the Shortness of Life or Marcus Aurelius' Meditations or the Incairidion of Epictetus to figure out that the Stoics did not fear death, but welcomed it just as you would welcome the change from adolescence to adulthood. I would like to share with you two quotes on the Stoics' view of death, one for Marcus Aurelius and the other, Seneca. Death would be a thing of dread if it could stay with you, but death must either not come at all or must come and pass away, Seneca. And for Marcus Aurelius, pass then through this little space of time comfortably to nature and end your journey in content, just as an olet falls off when it is ripe, blessing nature who produced it and thanking the tree on which it grew. I'm paraphrasing Seneca here, but is it not a great deed to conquer life and an even greater deed to conquer death? Please take my word for it that if you are struggling with anger, lack of motivation, fear of what others' opinions are of you or even wrestling with your own mortality after a brush with death or a family member's brush with death, Stoicism may be the guide that can make all the difference. You don't have to be a Stoic sage to find these teachings helpful, just willing to consider new ideas. I'm going to list a few books that will give you a good idea of what the Stoics believed. They are Seneca's Letters from a Stoic and On the Shortness of Life, Marcus Aurelius' Meditations and the Enchiridion of Epictetus. To purchase all of these books, it will only be approximately $30. For that $30, you will get your money back in wisdom tenfold. The link to all of these is in the description below. Thank you for taking the time to watch these videos of mine. If you're not already subscribed and did enjoy the video, consider doing so. These videos can only get better. Thanks for talking philosophy with me. Until next time.


Kamena Stalinka was founded in 2010 by Miguel Harpadoia, then Music Director of the Fort Worth Symphony Orchestra. Our mission is to discover, preserve, expand, and disseminate the rich musical legacy of the Americas through research, composition, publication, education, and performance. Our primary initiatives promote the growth of understanding and artistic ability and competency of young orchestral musicians and conductors. They are the Conducting Institute and the Texas Institute for Orchestral Studies. This summer, we welcomed 19 students for the Conducting Institute and seven young musicians for the Texas Institute for Orchestral Studies from across all of the United States, Central Europe, Asia, and South America. The Conducting Institute is a comprehensive program to teach, guide, and encourage the next generation of orchestral conductors and musical leaders. I could not possibly speak more highly about the Fort Worth Conducting Institution. Just the fact that there is a live professional orchestra that you're able to rehearse and work with is completely irreplaceable. And I have to say, Maestro Harpadoia has done such a phenomenal job in creating a very well-rounded program that not only just teaches you about the actual music knowledge and content that you need to know, but also what to do when you find yourself in a real-life situation. My experience at the Conducting Institute has been beyond what I imagined possible. The classes have all helped me approach my scores and given me a mechanics foundation that I needed. The podium time, conducting an appropriately sized orchestra for Beethoven or Brahms has been priceless. I love the Conducting Institute. I've been attending for four years now and I look forward to doing it again and again. Miguel Harpadoia is the best teacher you could possibly have. He has taught me so much and the participants students are amazing. We've formed such a great group and now I have friends, colleagues that I can talk to when I have questions. I love it. I love coming to the Conducting Institute because every time I work with Miguel Harpadoia, he broadens my awareness of my own conducting. So he teaches me to teach myself and my conducting doesn't just improve at the Institute but for a long time afterwards as well. My name is Wenning. I was a participant of the Conducting Institute in summer 2022. During the three weeks at the camp, I've learned so much on conducting by having ear training classes, observing rehearsals and of course standing on the podium. We believe that every conductor must have the fundamental knowledge and skills to study and understand a musical score and to lead an orchestra in rehearsing and performing that score. In its first season, the Texas Institute for orchestral studies successfully integrated its participating musicians in a three week side-by-side experience with the Conducting Institute laboratory orchestra's fellows and professional musicians. TX iOS was an overall fun experience but it was also really challenging at the same time and seeing the progress you make over time is really rewarding. TX iOS enriches and motivates young musicians through a rigorous orchestral immersion program which exposes participants to rich musical literature and incubates the talents and interests of pre-professional musicians by working closely with accomplished professionals in an intensive yet nurturing environment. Fort Worth's quality of life is enhanced by Kamino Stalinga's initiatives to increase the understanding of sophisticated new music of the Americas to deepen and broaden the reservoir of musical talent residing and working in the Fort Worth area and to stimulate achievement and appreciation by young school-aged musicians in ways that are not possible to provide in a regular school setting. These initiatives are complementary to and collaborative with the in-school curriculum. Our work reinforces and enhances the sophistication and cultural depth of the Fort Worth community. We believe this work is important to our society by enhancing the musical experiences of listeners and for the personal benefit of the participants who attain new and improved skills, self-confidence and a sense of personal achievement. They are then better equipped to share their talents and experiences with their friends, associates and communities at large. To learn more about Kamino Stalinga and our programs please visit KaminoStalinga.org


CHAPTER XIII. ON THE ROAD TO RUNNINGMEED. PART I. I was in Stamford in the year of the great charter of King John. Half the night's embarrass of all England, with a goodly following of men at arms and yeomanry, had been assembled under the banner of our stout marshal, Fitz Walter, and had seized by force and arms full many royal castles. Now, at the end of a truce, which to no avail had been secured by the archbishop, we were ready to march towards London to bring to terms our most crafty and tyrannic lord and king. For years he had dealt in plots and scheming to overreach the great and strong among the baronry, and from the weaker seized their lands and goods at will, and offed through their persons into endurance to further his gross ends of gain or vengeance. Now some hundreds of the barons of the north, with a dozen or more of us from the west counties and the Welsh marches, and a sprinkling of churchmen, who no less than ourselves had suffered from the king's aura reaching, were gathered in Bermancy Hall to agree, if we might, upon a scroll of the grievances that the king must remedy when our further assaults should have forced him to sue for peace. Geoffrey, lord of Carleton and Tarramore, leader of a hundred lances and half a thousand bowmen, rose from his seat amid a clamour of disputing voices, and saluted the marshal and the assembled company. "'I propose, my lords and gentlemen,' he said in that high, sweet voice of his which yet is far heard and commanding. The name of Sir Cedric de La Roche, knight of Grimsby and bold defender of our western marches, for the fifth and final member of this group. He is a brave man and true, and half, as we often say in the west, a head as well as an arm. He is both soldier and scholar forsooth, and knoweth more of the Latin tongue than any layman among us. You have named Sir Richard of Mountjoy to serve you in this matter, because, three months ago, he took the castle of Ternoy, which the king's men were strongly holding with greater forces than his own, and from whence they might have sorely threatened us. But most of you know not that was Cedric de La Roche, who gained entrance to the castle in disguise, and full well deceived the garrison, then at midnight overpowered, gagged, and bound the sentinel at a little poster-gate, threw it open, and admitted the Mountjoy's. Lacking him and his stratagem, we might still be hammering at the walls of Ternoy, and our whole campaign be sore delayed. For the Latin we have the Abbot of Moberly, said old Lord Esmond from his seat on one of the benches at the right. What need have we of another clerk? The reverend Abbot, answered Carlton, will do the cause good service, I doubt not, in making clear for our commissioners the substance of old scrolls and charters which they must study, and may have, in inditing in fair Latin, hand the articles which we present to the king. In his hands we may be sure the interests of his order, and particularly of the Abbot of Moberly, will not suffer. But I say, Tis well, that we of the baronage have a representative of our own number, who can see that this scroll for which we risk our lives and fortunes truly and amply provides for remedy of the wrongs we suffer. And I say, shouted Lord Esmond, springing to his feet the instant Carlton had finished, that if we are to have a representative of our order on the inditing of this scroll, as my Lord Carlton says, we should have a representative indeed. De La Roche is a true man and a capable soldier, as none will deny, but we have here many lords and gentlemen of longer service and of purest norm and blood. The night of Grimsby, as all may know, is Yeoman and Saxon born. Such a man, be he never so learned, must ever think as the folk from whom he sprung, and can never rightly guard our rights and privileges. For an hour we had debated of our wrongs and the measures that should put an end to them, each speaker being fiercely bent upon the thing that should lift the oppression that had borne most heavily upon him, and caring little for ought else. But finally it was seen that the whole assembly could accomplish not but argument and loud bickering, and that the writing of the scroll must be done by a few chosen men who should later bring their work before the whole body of leaders for their ascent and undertaking. Two of the oldest of the northern leaders, the Baron de Longville and the Lord of Esmond, had been first named. Then the learned and courteous-like Abbot of Moberly, who was beneath the insurgent banner because of the king's high-handed procedure in the matter of Moberly Abbey, where, during the absence on pilgrimage of the rightful holder, he had declared the apacy vacant and conferred it with all its lands upon one of his shameless favorites from Normandy. A moment before my own name had been added to the list in recognition of the services of the western lords that had well broken the power of the king in all their countryside. Following Lord Esmond's bitter speech came shouts of approval from some of the other northerners, and it seemed like my old friend and comrade would be deprived of the honor which Geoffrey of Carleton had sought to have conferred upon him. But the venerable de Lacey, long the Lord High Constable of England, and still a power in the land, though bent and snowy-haired with age, rose slowly to his feet and addressed the marshal and the company. My lords, tis well for those to talk who know whereof they speak. Years agon I knighted Cedric de La Roche for nightliest service at the battle of the past, where verily he changed defeat to victory. Since that time he hath many a time, and oft, served under me and others, always to the welfare of the kingdom and the enhancement of his name. Lord Esmond says that Cedric de La Roche comes not of noble family. I ask of you, my lords, who made our family's noble but some hard-smiting ancestors we had that served not better I warrant you than this man of whom we speak? And I have seen his lands of Grimsby and the stout and loyal men who do willingly follow him, and know full well he can think and plan as well as strike. Finally, my lords, tis not the tale of his fathers or his grandfather's deeds but of his very own that should guide the choosing of a man for a time of need. At this still louder shouts burst forth, especially from the younger men, and some did loudly call Sir Cedric's name, insisting that he serve. When partial silence came once more, the marshal brought all questioned to an end by announcing all the names of the group and ending with that of Cedric de La Roche. Then, it being near the supper-hour, the company broke up amid cheering and noisy overthrow of benches and the clamor of many voices in eager talk of the day's events. The meeting next day of the group that should do the writing of the scroll was scarcely better than that of the whole assembly. Esmond and De Longville disputed long and loud over exemption from the tax levied for the French war, and some suggestions that we others made for the kingdom's better ordering went all unheeded in the din. The abbot, smiling and crafty as always, patiently awaited the time so sure to come when noise and clamor should exhaust itself, and his own smooth-spoken counsel should prevail. He had with him a copy of the old charter of the First Henry, and Cedric a draft of some of the laws of Edward the Confessor, which he believed should be included. At last, when twas seen that we made no headway, my own voice was for a moment listened to, and twas agreed that our two scholars, the abbot and Cedric de La Roche, should work together, making from the ancient laws and grants, with such additions as were found needful, the articles we should put before the king. With all my comradely thought for Cedric, I could but smile as I thought of the task that now confronted him. I knew well that he had certain cherished plans with regard to these articles, whereby he hoped to gain for the commons some of the privileges and immunities which he regarded as the natural rights of free-born men. Often and often he had declaimed to me of these things, and with such eloquence and conviction as well nigh made me a convert to his party, if that could be called a party which had no leaders, and no program, and scarce a voice save his own. The commons knew no other way of protest against the wrongs they suffered than such violent and fruitless revolts as that of the churls of delancy manner, with mayhap the killing of a tyrannous noble, and the later hunting down and hanging of the leaders of the mob. Cedric had for years maintained that their natural rights should be assured to them by Charger, and not left to the caprice of some careless or greedy overlord. But the abbot of Moverly was allied by blood and by early training to powerful Norman families, and was likely that he had but little sympathy with any such ideas. Handsome, learned, and eloquent, he was accustomed to win his way among rough and heavy-handed lords and barons, and the little better schooled officials of the royal courts by the skill and grace of his address, and yet more, if all rumors were true, by a readiness to shift his allegiance to any cause in accordance with circumstance and his own prevailing interest. In truth he had been bred for the law as much as for the church, and his great services to his order, which had been amply rewarded with power and place were those performed in court or council rather than in church or monastery. At this very time Lord Geoffrey of Carlton, Cedric, and I had reason to suspect the abbot of secret communications with the Archbishop, who was still nominally of the king's party, and who would perhaps have much to do with the final shaping of our articles if ever we should force the king to consent to their sealing. It was evident that the rights of churchmen would not be overlooked in the final treaty, and although this too had our approval, we were the more determined that those of other estates should also be well guarded. On the morrow, nevertheless, it seems certain that this co-working of two such diverse men would be effective, and that we would soon be prepared to take before the assemblage of leaders the completed scroll. The abbot and Cedric de La Roche came late to our meeting and still debating hotly on the way, but they brought a list of articles they had most cunningly devised for the remedy of the ills of which we most loudly complained. The abbot read them to us clearly and with most just accent like the learned speaker that he is, and I think the two old northern lords were modally impressed with the power and worth of words so skillfully marshaled. When he had finished, we might have then and there adopted the articles and ended our labours. But at the end of his reading the abbot said, my lords, I wish to testify that from Cedric de La Roche I have received most welcome assistance in the drawing of this scroll, both in the reading of the ancient laws and chargers, and in the devising of new provisions toward the wise and just ordering of the kingdom. Nevertheless, upon some minor points we have not yet agreed, and upon these he wishes to address you. Sir Cedric rose to his feet and for a moment looked from one to the other of our company. His fine and open countenance and clear blue eyes and the marshal squareness of his broad shoulders would have won him high regard in any great assembly. It seemed to me at that moment that the youth whom I had first known as the Forrester of Pelham and whom I had seen rise to nightly dignities, well deserved, was at the summit of his career when those whose decisions were weighty in the affairs of our time awaited his words on a matter of such moment. Baron de Longville was looking at Cedric with no unfriendly eye, but the lord of Esmond, who had wished to adopt the articles at once, frowned with impatience at the end of the abbot's speech, and now gazed moodily at the floor. My lords began Cedric clearly. We have, as the twentieth of these articles, let no sheriff or bailiff of the king take horses or carts of any free man for doing carriage except with his own consent. Upon the next page we have the provision. Let not the body of a baron, knight, or other noble person be taken or imprisoned or deceased or outlawed or banished or in any way destroyed, nor let the king go or send upon him by force except by the judgment of his peers or by the law of the land. These things are just and right, but to my thinking they go not far enough. Why should we not deserve the good wishes for the triumph of our cause and the strong right arms not only of the baronage but of all the free men of England? Why should not these provisions be altered to guard their rights also? Lord Esmond raised his head and gazed sharply at Cedric's face. And how wouldst thou amend them? he growled. I would say in the first instance, let no sheriff or bailiff of the king nor any other person take horses or carts of any free man for doing carriage except with his own consent. And in the second would have the words, a free man in place of baron, knight, or other noble person, so that it would read, let not the body of a free man be taken or imprisoned or deceased or outlawed, and the rest. May have these churls have made thee their spokesmen, sneered Esmond. Nay, replied Cedric, I speak for no party, whether high or low, but for the common good of England. Lord Esmond turned with sour and vinegary look, first to the Longville, then to the Abbott. What did I say in the assembly? This man hath no conception of the rights of our order. All his concern is for churls and clowns. Cedric grew very red, and his hand went to his sword-hilt. I sprang up to address our chief, the Longville, and placed myself between the night of Grimsby and the fiery old lord from the north. My lords! I cried. We gained nothing by arguments that speedily pass into brawls. Come, let us vote upon these provisions. Tis the rightful way. Tomorrow, or the next day at the furthest, we must take our report to the assembly, and we should come to agreement. Tis so, replied to Longville, we waste our time in bickering. Come, Esmond, what say thou as to these amendments? I say nay, shouted Esmond. Let the articles even stand as they were. And thou, most reverent Abbott, I say nay, replied the churchmen quietly. And thou, Mount Joy? I, I answered loudly. These changes seem to me to take not from us and to be well-conceived to gain us many friends. De La Roche? I. Longville gazed first at the floor beneath his feet, then at the ceiling overhead, and bent his brows in a painful frown. At length he said, It seems I have the casting vote. I see little use in these changes, saved to pamper churls and thralls, that too often already raise their heads with complaints and demands. Some of them verily believe they might govern the land as well as their betters. Tis a dangerous tendency that must be checked. I say nay, also. Lord Esmond turned toward Cedric with a smile of triumph, and my heart became as led to think of his defeat. But the night of Grimsby was instantly on his feet again with a new proposal, which to my amaze he uttered with a broad and pleasant smile on his face, such as he might have warned had his amendings been received with utmost acclaim. Has the thought come to you, my lords, that in this scroll, thus far, we have made no provision for the enforcement of our demands? We deal with a strong and crafty monarch, even if he places seal upon our demands, what surety have we that he will adhere to them after our levies have been dispersed? He will then be stronger than any one or two or three of us. How shall we ensure his adherence to the treaty? The rest of us gazed at one another in silence. This was a new thought, it seemed, to our whole assembly, and none could deny the seriousness of the question. At last DeLongville spoke again, and hast thou, Grimsby, given thought to this so that thou canst now produce a remedy? Not on the instant, my lord, but in the main my thought is this. In this instrument itself must be provision for its enforcement. The king must agree that a body of ten or a score or more of us shall be named by ourselves, and that these shall be responsible to see that the charter be not impaired or overridden. In another night I can form the language to carry this provision into our articles. Then the abbot spoke, suggesting that Sir Cedric be instructed to do this, and finally on motion of mine the articles were back referred to Cedric and the abbot, with instruction to bring to our meeting at two of the clock on the following day a fair and perfect copy that we might adopt and place before the assembled leaders. CHAPTER XIII On the road to Runnymede, Part II T'was then high noon, as we left the council-hall, Sir Cedric took me by the arm and insisted that I come to his inn for the midday meal. There was in his inviting a special urgency, and a look in his eyes from which I, who knew him so well of old, instantly gained the knowledge that this was no ordinary matter of courtesy, but something a vastly greater moment. So I easily suffered myself to be led toward his quarters, and soon we receded at a board that was graced with a goodly roast and all other due refreshment. When we had something satisfied our hunger and the old serving man who waited on us had departed, Cedric bent toward me across the board to say, What sayest thou Richard, to a ride of a dozen leagues or so, and a little adventurer, whereby, if fortune favors, we may do our cause full loyal service? With all my heart, I cried, Whither shall we ride and on what errand? T'was two months and more since we had seen activity, and this dull life of the camp and the town was little to my liking. Sir Hubert Gillespie had lately struck a blow for the king by the surprise and capture of two strong castles in the Midlands that we had thought safely in our hands while we, with our brave array at Stamford, consumed the days and our dwindling substance in idleness. "'Tis one that's something dangerous, forsooth,' replied my friend, and I doubt much whether our elderly and prudent leaders would approve. Say no more, for Mount Joy is with thee to the hilt. What followers shall I bring, and with what arms? A dozen lusty swordsmen, men still young and light on the feet and with heads to understand a stratagem. Dickon and John of the Wallfield and Elbert the Smith are the right sort. See that every man wears beneath his outer garment a coat of linked mail and carries a sword no longer than his arm. Within the hour I will meet thee at the beech wood thou knowest to the south of the town, and will bring a like number of the men of Grimsby. We shall ride hard and far, so look to it, I pray thee, that thy men be well mounted. We may have cause for speed on the homeward road.' An hour later, with four and twenty proper men, Cedric and I rode out of the beech wood and took the high road toward the south, where, but five or six leagues away, the castles and most of the towns were still in the hands of the king's mercenaries. I knew full well that the quest on which we were embarked was one that meant our cause's advancement and would have willingly trusted Cedric for the rest. But now we drew ahead of our horsemen, and he explained full clearly his design. T'was such a plan as only Cedric would have formed, and his outcome in truth exceedingly dubious. But we were comrades of old in many a venture that would have been refused by prudent men, and now he had no labour in convincing me that this was worth the trial. After an hour's riding we came to a thick wood, and turned aside in this into a little glade where we halted to rest our mounts and to bring about a most surprising change in our apparelling. At a word from Cedric, each of the Grimsby men proceeded to withdraw from his saddlebags some garments which, being unfolded, appeared as the long gray cloaks and hoods of palmers. Each, it seemed, had brought a costume for himself and for one of the Mount Joy men. And now, in less time than the telling takes, we had all laid aside among the bracken any headwear or other dress that might not properly consort with these, and stood forth as a body of pilgrims in the dress that marked those who had accomplished the toilsome journey to the Holy Land. Soon we were on the road again, and, save for now and then the rattle of a sword-hilt or a robust laughing-word, might not have been distinguished from a cavalcade of devout returning pilgrims such as were not uncommon on our roads. Without mishap we pursued our way into a region where all the points of advantage were held by our enemies, and where armed parties far too strong for our gain-saying patrolled the roads or watched them from the hill-tops. In the late afternoon we came with insight of the Castle of Moberly, which was held for the King by Sir John Champney, with a hundred lances and six score cross-bowmen. On the left, and but half a mile from the castle, lay the Abbey where William de Belair, favorite of the King, renegade cleric and foresworn crusader, held usurping sway over the monks and lay-brothern, and the fields and vineyards that had been the rightful domain of our associate at Stamford, whom we still greeted as the Abbot of Moberly. At a light distance from Moberly Castle was a fork in the road just beyond a timbered bridge or a stream. There the left-hand track led to the Abbey, and that on the right went straight to the castle gates. At the full trot we took the former turning, and soon were calling for admittance at the Abbey doors. This, to a devoted band of pilgrims, was not long denied. The gates were thrown ajar, and, leaving two trusty fellows to care for the horses in the outer courtyard, we passed into the Reflection Hall of the monastery to pay our respects to this venerable seat of piety and learning. Our worthy palmers scattered themselves about the great room with its timbered ceiling and mighty fireplace, and engaged in talk with the monks, or in reverent examining of the painted series on the walls, the work of an earnest, though not too highly skilled, lay-brother, and setting forth the story of Joseph and his brethren. After a little, Sir Cedric, acting as our leader, sent word to the Abbot whom we had not yet seen, that here was a group of a score and more of palmers who now paid their first visit to the far-renowned Abbey of Moberly, and who wished to have speech with the reverent master of the house ere they departed. This message, with its accompanying compliments, accomplished its intent, and soon William de Belaire, in all the robes of his office, entered the hall from an inner door and seated himself in his great chair on the days. If ever the character and history of a man were written on his face, it was so with the false Abbot of Moberly. My gorge rose within me at the sight of his red and bloated countenance that told so plainly of a life the very opposite of that led by a true monk and churchman. His mean and shifty little gray eyes were all but covered with folds and wrinkles of fat, yet quite sufficiently revealed a nature compounded of fox and pig. de Belaire was one of a group of dissolute Frenchmen who had won the favour of the king and the hatred of true Englishmen by supporting our lawless and grasping sovereign in all his schemes for the seizure of power and wealth. It was against them nearly as much as the king that our banner of revolt had been raised, and in our articles of Stamford we had already named a half dozen of the worst of them, who must be deprived of all offices and banished from the kingdom. To his no blame to the church that such miscreants profaned some of her holy offices, in defiance of her rites of ancient usage they had been thrust by the royal master into the places they disgraced, often times in reward for services which would not bear recording. Reverend Father, said Cedric Bowinglow, we congratulate ourselves upon our visit to this ancient and honourable Abbey, and we have here some gifts and tokens to bestow upon thee as the head of this worthy brotherhood. de Belaire bowed deeply in acknowledgement of this greeting. When he raised his head again, what was his amaze and horror to find that he that had addressed him so respectfully had sprung upon the days, pulled from his shoulders the Palmer's cloak, and now rushed upon him as a hound upon his quarry. In an instant the long grey robe was flung or the abbot's head and arms, and despite his struggles and cries, a rope was speedily bound about his middle, pinioning his hands to his sides. Then he was lifted bodily and hurried toward the courtyard door. Some of the monks set up a hideous outcry, and one or two sought to intercept those who carried the bound and struggling abbot, but where they thought to deal with unarmed pilgrims, they found themselves confronted with two and twenty stout fellows, each of whom had drawn from beneath his flowing cloak a short-bladed sword and flourished it in most menacing way. They fell back before us over odd, and understanding nothing of what had passed, only one of the monastery people did preserve his wits at this amazing juncture, and this an acolyte youth of sixteen years. Slipping out of the hall and through the rear of the abbey, he ran, as we afterwards learned to our cost, with might and main to take the news of this mad foray to the castle's governor. In the outer yard we spent some time in adjusting more firmly our captive's bonds and in cutting slits through the cloak that bound his head so as to allow him to breathe, but no eyes to see, and scarcely to make himself heard with calls for help. Then hoisting him with difficulty, for he was a gross, fat man. Upon a stout charger whereon one of our own men rode behind him, we turned away from the abbey, and rode at such speed as we might on the road by which we came. Our progress was slow at the first, for our prisoner sat most unevenly in his bonds, and we had no mind to let him fall by the way. And we had no more than fairly set out on the road when he began to shout and halloe, in such wise that Dickon of the Wallfield, who rode behind him, was feigned to bring him to understanding of his hopeless plight by a sharp prick from his Ponyard's point. Thereafter he was silent, and we made better way, but with all most precious time had been lost. The night had already fallen, and with another quarter hour we might have won safely away. But as we approached the fork of the road, we heard a thunder of hooves coming from the castle. The riders were nearer the joining than we, and ere we could gain the bridge, we heard their horses upon it, and knew that Sir John Champney's men were drawing up in battle array to meet us. As we surmised even then, Sir John had divided the force that he so hastily summoned to punish the supposed outlaws who seized the abbot for a ransom, and had sent one party straight to the abbot, and led the other to this point to intercept us. In the light from the great moon now rising, we could see that their numbers were more than twice our own. They were variously armed as was to be expected with men who had been so abruptly summoned forth, but there were lances and steel capsineaux, and some had coats of mail. We sorely wished for the good broadswords we left behind at Stamford, or the crossbows with which a dozen of our party were so skilled. But now was not time for hesitation or for choosing of courses. Well we knew that in a trice the other party, riding from the abby gates, would be on our track, and we would be taken in front and rear. With a mighty shout we rode down upon the bridge, trusting all to the darkness and the fury of our attack. In a moment we were in the midst of a bloody melee on the bridge. Our men thrust back their hampering robes and hewed and slashed with deadly effect, but those opposing us were no weaklings nor novices in war. Sir John Champney slew two of our men with downright broadsword strokes, and another was pierced through throat by a lance. I rode in a closer press of fighting than I had seen since the battle of the past, and once or twice was near beaten from my horse, though some of those that reigned their blows on me fared worse indeed. Then Cedric came face to face with Sir John Champney, received a broadsword stroke on his uplifted, male-clad arm, and countered with a blow that sent his enemy to earth. Instantly the cry arose that Sir John was slain. Most of his followers were French and Flemish mercenaries, and now they melted away before us, fleeing to the fields on either side of the bridge, or leaping to the shallow waters below. We paused long enough to learn that our men who had fallen were past all help, then rode forward at a gallop up the moon-lighted way, with our prisoners still safely bound and in our midst. By the eleventh hour we entered again the wood where we had transformed ourselves to palmers, and towards the work of but a moment to change us back to knights and men at arms. By midnight we were safely in the town, and had our prisoner properly bestowed. Then Cedric and I parted for the night, I to go to my bed, and he, as the morrow showed, to labor by candlelight all through the hours of darkness. At nine the next morning I was by appointment at Cedric's lodging, and found that he had just dispatched a messenger to the true Abbot of Moberly, with an urgent request that he come at once since most important news awaited him from the Abbey itself. This message speedily accomplished its object, and the Abbot, standing not on ceremony, came hurrying to the lodgings. We greeted him most courteously, and when our guest was duly and comfortably seated, Cedric stated that riders had come in from Moberly the night before with the news of a most surprising happening. A band of a score or more of pilgrims returning from the Holy Land had entered the Abbey, and doubtless being Roth at William de Belair, because he had foresworn himself by abandoning his vow to go on crusade for the recovery of the Holy Sepulchre, had seized and bound him, and overawing the monastery with weapons, had carried him away by force. The Abbot listened to this tale of violence with sparkling eyes, and with no hint of censure for those who had so roughly laid hands upon a cleric dignitary. When it was finished, indeed, he could scarce restrain his glee. Rising and smiting the table roundly with his hand, he cried, Ha! well served, well served indeed, for a creature that calls himself monk and abbot, for sooth, when profit is that way to be gained, but who forgets all monkish obligations when a layman's way of living better serves him. The palmers are right, indeed, and I devoutly hope they may keep him, for I, as far from Moberly Abbey, as his conduct had ever been from that of a true churchman. Cedric then resumed, in slow and measured voice. It so happens, Reverend Abbot, that I have several friends among these palmers, and to some extent they rely on me for advice in this matter. Ah! is it so, indeed, questioned the Abbot eagerly. Then I trust that thou, as a true friend of the church and her rightful servitors, hast given advice to hold this fellow they had taken, at least till the king be brought to terms, and our brotherhoods be free again to fill their offices without dictation.  Nay, my advice has not yet been given, till require some further meditation to be sure that his wisely bestowed. But Reverend Abbot, if thou wilt but climb the stair that I shall show thee here, and apply thine eye to a hole in the wall at the right, near the top, I warrant thee a sight well worth thy pains. So saying, Cedric rose, and throwing open a small door at the rear of the room, indicated a dim and curving staircase that rose beyond it. The Abbot, after a searching glance at his host, as though he feared some stratagem, quickly mounted, looking eagerly the while for the eye-hole in the wall. Both of us remained below, and Cedric, turning to a cabinet, withdrew from it and placed upon the table a huge scroll of many sheets of freshly written parchment. A moment later the churchmen returned with brightly glowing face and twinkling eyes, and when the stairway door was closed again exclaimed, Sir Cedric de la Roche, thou art a true friend to the church, and thy services shall be well remembered. Tis William de Belair, beyond all doubt, who sits in yonder inner room, and his two archers of Grimsby who guard him. Full well do I know who led that band of pommers, and I say again, thy fortunes shall not suffer for it. Cedric bowed and smiled. Ah, well, tis neither here nor there who led the pommers, or whether they acted wholly of their own impulse. The thing of greatest moment now is this scroll of the articles which I have here in fair copy. Read it, I pray thee, and see whether thou wilt give thy voice for its adoption. Thou wilt see that I have introduced the provision for five and twenty barons who shall enforce the charter, and also have written in some other matters that seem to us of moment. The abbot took the scroll and quickly conned the pages whereon he and Cedric had on the first day of their labours come to full agreement. Then he came to the twentieth article, and, ceasing reading, looked up at Cedric sharply. Thou hast here the wording for which thou didst argue yesterday. I, tis so, answered Cedric grimly. Read on. The abbot complied, but quickly came to another stop. Let not the body of a free man be taken or imprisoned, he read. That again is the very language that was yesterday rejected. Cedric nodded innocent. Read on, he said. For some pages the abbot went on in silence. Then he uttered an exclamation of surprise, and paused to read again, this time allowed, an article that appeared near the end of the scroll. All the aforesaid customs and liberties, which the king hath conceded, to be held in the kingdom as far as concerns his relations to his men, all in the realm, as well ecclesiastics as laity, shall on their part observe toward their men. The abbot leaped to his feet, his face red with wrath. What means this, Delaroche, would thou have all these things for which we risk our lives and lands extended to every churl and varlotte in the kingdom? I, answered Cedric subtly, and if thou look abroad through our camp, thou see some thousands of those same churls and yeoman that do risk their lives in this cause as much as thou or me. The abbot shook his head with impatience. Tis beyond reason, Delaroche, I cannot give my word for it. Cedric for a moment gazed out of the window. Then he said to me, this keeping endurance of an ecclesiastic who was appointed to his place by the king, and more overstands high in his favor, is a difficult and dangerous business. T'will be better if we take him to the town's edge and turn him loose to find his way back once he came. The abbot gazed at Cedric with parted lips and baited breath while one might have told to score. Then, of a sudden, he flung the parchment on the table and laughed, full, loud, and long. Thou hast won, Delaroche. I yield me. Thou hast won, and fairly. Thou art a most persuading speaker. I'll be bound. I will go before our group this day and make them adopt these articles, whether they will or no. Then tomorrow I will speak for them before the whole assembly. Thou shalt see what I can do when I am well put to it. Depend upon it, the articles of that very scroll that lies before us, will be the ones our party will present to the king. And thou, on thy part, shall have dew, watch, and ward kept of thy prisoner, and see to it that he by no means gains his liberty until the king hath sealed our charter, and pledged himself to interfere no more in our clerical elections. The abbot was as good as his word. That afternoon he delivered such an address and eulogy of the articles, as they appeared in this latest scroll, as I had never heard before on any subject whatsoever. He marshaled all the arguments Cedric had used, together with many more he had not thought on. His speech was filled with grace and eloquence, and was of an enthusiasm that carried all away. He showed, beyond all doubt, the power that would accrue to our party through this inclusion of the rites of the commonality in our charter. When he was done, DeLongville as strongly favored these provisions in the articles, as on the day before he had opposed them. Lord Esmond grimly held his peace, though oft shaking his gray head in denial, and soon the scroll had been adopted by our vote of four to one. The following day our ardent champion made a yet more eloquent speech before the full assembly, and the articles were approved by acclamation. All know the remainder of the tale of Magna Carta, how the king, three days later at Brackley, where the articles were read to him, refused them with an oath, furiously declaring that the barons might as well have asked of him his kingdom, how we resumed the war forthwith, and the taking of his castles, how the gates of London were open to us, and the king was at length brought to terms at Runnymede. There again was Cedric de La Roche, and the Appet of Moberly, who conferred with the Archbishop and the other commissioners of the king, and satisfied themselves and us that the completed scroll that received the royal seal was to the same effect as our articles of Stamford and Brackley. And now King John is dead, and little lamented, and a wiser sovereign rules the land. Already men begin to see how great a thing was done at Runnymede. Cedric said that the great charter will be for centuries to come the basis of our English law, since it affirms with equal voice the rights of all our three estates, the nobility, the clergy, and the commons. It seems to me that later generations will find in its provisions the authority and the suggestion for many a reform that we dare not yet attempt, and that freer and happier men may date the beginning of better things to our bitter struggle with King John. If so be, may they think not over much of us that were noble-born and fought for lordly privilege, but may they never forget that in our day there were true men of lowly birth who risked their all for the rights of their fellows. Of these was none more worthy of honour than he whom I am ever proud to call my friend and comrade Cedric the Forrester of Pelham. THE END