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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:
 Ordinary Differential Equations Exercise 1
Imports
Step2:  Euler's method
Euler's method is the simplest numerical approach for solving a first order ODE numerically. Given the differential equation
$$ \frac{dy}{dx} = f(y(x), x) $$
with the initial condition
Step4:  The midpoint method is another numerical method for solving the above differential equation. In general it is more accurate than the Euler method. It uses the update equation
Step6:  You are now going to solve the following differential equation
Step7:  In the following cell you are going to solve the above ODE using four different algorithms
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Python Code:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
from scipy.integrate import odeint
from IPython.html.widgets import interact, fixed
import math
Explanation: Ordinary Differential Equations Exercise 1
Imports
End of explanation
def solve_euler(derivs, y0, x):
    Solve a 1d ODE using Euler's method.
    
    Parameters
    ----------
    derivs : function
        The derivative of the diff-eq with the signature deriv(y,x) where
        y and x are floats.
    y0 : float
        The initial condition y[0] = y(x[0]).
    x : np.ndarray, list, tuple
        The array of times at which of solve the diff-eq.
    
    Returns
    -------
    y : np.ndarray
        Array of solutions y[i] = y(x[i])
    
    h = x[1]-x[0]
    y = [y0]
    for i in range(len(x)-1):
        y.append(y[i] + h*derivs(y[i],x[i]) )
    return np.array(y)
solve_euler(lambda y,x: 1, 0, [0,1,2])
assert np.allclose(solve_euler(lambda y, x: 1, 0, [0,1,2]), [0,1,2])
Explanation: Euler's method
Euler's method is the simplest numerical approach for solving a first order ODE numerically. Given the differential equation
$$ \frac{dy}{dx} = f(y(x), x) $$
with the initial condition:
$$ y(x_0)=y_0 $$
Euler's method performs updates using the equations:
$$ y_{n+1} = y_n + h f(y_n,x_n) $$
$$ h = x_{n+1} - x_n $$
Write a function solve_euler that implements the Euler method for a 1d ODE and follows the specification described in the docstring:
End of explanation
def solve_midpoint(derivs, y0, x):
    Solve a 1d ODE using the Midpoint method.
    
    Parameters
    ----------
    derivs : function
        The derivative of the diff-eq with the signature deriv(y,x) where y
        and x are floats.
    y0 : float
        The initial condition y[0] = y(x[0]).
    x : np.ndarray, list, tuple
        The array of times at which of solve the diff-eq.
    
    Returns
    -------
    y : np.ndarray
        Array of solutions y[i] = y(x[i])
    
    h = x[1]-x[0]
    y = [y0]
    for i in range(len(x)-1):
        y.append(y[i] + h*derivs( (y[i] + h/2 *derivs(y[i],x[i])) , (x[i] + h/2)))
    return np.array(y)
assert np.allclose(solve_midpoint(lambda y, x: 1, 0, [0,1,2]), [0,1,2])
Explanation: The midpoint method is another numerical method for solving the above differential equation. In general it is more accurate than the Euler method. It uses the update equation:
$$ y_{n+1} = y_n + h f\left(y_n+\frac{h}{2}f(y_n,x_n),x_n+\frac{h}{2}\right) $$
Write a function solve_midpoint that implements the midpoint method for a 1d ODE and follows the specification described in the docstring:
End of explanation
def solve_exact(x):
    compute the exact solution to dy/dx = x + 2y.
    
    Parameters
    ----------
    x : np.ndarray
        Array of x values to compute the solution at.
    
    Returns
    -------
    y : np.ndarray
        Array of solutions at y[i] = y(x[i]).
    
    h = x[1]-x[0]
    y = []
    for i in range(len(x)):
        y.append(0.25 * np.exp(2* x[i]) - 0.5 * x[i] - 0.25)
    return np.array(y)
solve_exact(np.array([0,1,2]))
assert np.allclose(solve_exact(np.array([0,1,2])),np.array([0., 1.09726402, 12.39953751]))
Explanation: You are now going to solve the following differential equation:
$$
\frac{dy}{dx} = x + 2y
$$
which has the analytical solution:
$$
y(x) = 0.25 e^{2x} - 0.5 x - 0.25
$$
First, write a solve_exact function that compute the exact solution and follows the specification described in the docstring:
End of explanation
x = np.linspace(0,1,11)
h = x[1] - x[0]
derivs = lambda y,x : x + 2*y
plt.plot(x, solve_euler(derivs, 0, x), label="Euler's Method", color="yellow")
plt.plot(x, solve_midpoint(derivs, 0, x), label ="Midpoint Method", color="blue")
plt.plot(x, solve_exact(x), label="Exact", color="red")
plt.plot(x, odeint(derivs, 0, x), label="ODEint", color="green")
plt.xlabel("x")
plt.ylabel("y(x)")
plt.title("Methods of Solving ODEs")
plt.legend()
ax = plt.gca()
ax.set_axis_bgcolor("#fcfcfc")
assert True # leave this for grading the plots
Explanation: In the following cell you are going to solve the above ODE using four different algorithms:
Euler's method
Midpoint method
odeint
Exact
Here are the details:
Generate an array of x values with $N=11$ points over the interval $[0,1]$ ($h=0.1$).
Define the derivs function for the above differential equation.
Using the solve_euler, solve_midpoint, odeint and solve_exact functions to compute
  the solutions using the 4 approaches.
Visualize the solutions on a sigle figure with two subplots:
Plot the $y(x)$ versus $x$ for each of the 4 approaches.
Plot $\left|y(x)-y_{exact}(x)\right|$ versus $x$ for each of the 3 numerical approaches.
Your visualization should have legends, labeled axes, titles and be customized for beauty and effectiveness.
While your final plot will use $N=10$ points, first try making $N$ larger and smaller to see how that affects the errors of the different approaches.
End of explanation
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