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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 This is a demo of symbolic computation using Python and Sympy.
Step1:  We first try to solve a complicated problem
Given
Step2:  Lets transform the left hand side in teh same way
$$ y $$
Take the natural log
$$ ln (y)$$
Take the derivative
$$ \frac{1}{y}dy $$
Now the complete equation is as follows
$$
\frac{1}{y} dy = \frac{3x^2 + 15x - 8}{2x(3x^2 + x -2)}
$$
The right hand side is given above, so to find the solution just multiply $ y $
Step3:  We can simplify this as well and just take the derivative directly
Step4:  This was a round about way to do something simple, but we can compare them directly to make sure
Step5:  Or we can plot it
Step6:  Can make a substitution to evaluate the function
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Python Code:
import numpy as np
import sympy
sympy.init_printing() # pretty printing
Explanation: This is a demo of symbolic computation using Python and Sympy.
End of explanation
# must define symbolic variables before us
x, y = sympy.symbols('x y', real=True)
y = x**2*sympy.sqrt(3*x-2)/(x+1)**2
y # This causes it to output formatted nicely. 
# Taking the natural log of this:
a = sympy.ln(x**2*sympy.sqrt(3*x-2)/(x+1)**2)
a # Keep in mind that the left hand side is ln(y)
# Taking its derivative with respect to x
# So, we took the log of the right side, then the derivative
sympy.diff(a,x)
# That is kind of messy. Let the computer simplify it
sympy.simplify(sympy.diff(a,x))
# We could have done this all at once
sympy.simplify(sympy.diff(sympy.ln(y),x))
Explanation: We first try to solve a complicated problem
Given:
$$
y = \frac{x^2\sqrt{3x-2}}{(x+1)^2}
$$
Find: $ \dot{y} $
Now we'll define symbolic variables
End of explanation
sympy.simplify(sympy.diff(sympy.ln(y), x)*y)
Explanation: Lets transform the left hand side in teh same way
$$ y $$
Take the natural log
$$ ln (y)$$
Take the derivative
$$ \frac{1}{y}dy $$
Now the complete equation is as follows
$$
\frac{1}{y} dy = \frac{3x^2 + 15x - 8}{2x(3x^2 + x -2)}
$$
The right hand side is given above, so to find the solution just multiply $ y $
End of explanation
sympy.simplify(sympy.diff(y, x))
Explanation: We can simplify this as well and just take the derivative directly
End of explanation
sympy.simplify(sympy.diff(y, x)) == sympy.simplify(sympy.diff(sympy.ln(y),x)*y)
Explanation: This was a round about way to do something simple, but we can compare them directly to make sure
End of explanation
sympy.plot((sympy.diff(y, x), (x, 0.2, 10)), (y, (x, 0.5, 10)))
# To change colors
# show = False delays the plot until we can set all of the parameters
# legend turns on the legend and uses the labels we have later.
p = sympy.plot((sympy.diff(y,x),(x,0.2,10)), (y, (x,0.5,10)), show = False, legend = True)
p[0].line_color = 'blue'
p[0].label = '$\\frac{dy}{dx}$'
p[1].line_color = 'green'
p[1].label = '$y$'
p.show()
Explanation: Or we can plot it
End of explanation
from sympy.abc import alpha
y.subs(x,alpha)
Explanation: Can make a substitution to evaluate the function
End of explanation
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