The eigenvalues associated with the equation below are all nonnegative.y" + λy = 0;_y'(−π) = 0, y'(0) = 0First determine whether λ = 0 is an eigenvalue.Then find the positive eigenvalues and associated eigenfunctions.Plot the eigenfunctions, corresponding to n = 1, 2, 3, 4, 5.

As per the question, we are asked with exploring the eigenvalues and eigenfunctions of a second-order linear differential equation given bywith specific boundary conditions and .Our objectives are to:1. Determine if is an eigenvalue of this equation.2. Find the positive eigenvalues and their associated eigenfunctions.3. Plot the eigenfunctions for the cases where , corresponding to the positive eigenvalues.To determine if is an eigenvalue of the differential equationwith boundary conditions and ,we first substitute into the equation, which simplifies it to1. Solve the Differential Equation: The general solution of is a linear function: where and are constants.2. Apply the First Boundary Condition: The first boundary condition is . Since , the derivative is . Applying the boundary condition: This implies that , so the solution simplifies to , a constant function.3. Apply the Second Boundary Condition: The second boundary condition is . However, since , for all , so this boundary condition is automatically satisfied.4. Determine if is an Eigenvalue: For to be an eigenvalue, there must exist a nontrivial solution (i.e., a solution other than the zero function) that satisfies the differential equation and the boundary conditions. In our case, any constant provides a nontrivial solution.Therefore, is indeed an eigenvalue of the given differential equation with the specified boundary conditions.To find the positive eigenvalues and associated eigenfunctions for the differential equationwith boundary conditions and ,we need to consider . Let's denote where . The differential equation becomes1. Solve the Differential Equation: The general solution for this equation is where and are constants.2. Apply the First Boundary Condition: The first boundary condition is . The derivative of is Applying the boundary condition: This simplifies to 3. Apply the Second Boundary Condition: The second boundary condition is . Applying this to the derivative: Since , this implies .4. Determine the Eigenvalues and Eigenfunctions: With , the solution simplifies to . The remaining condition from the first boundary condition is This equation is satisfied for where is an integer. This gives for . However, since , we exclude . Thus, the positive eigenvalues are for .5. Eigenfunctions: The associated eigenfunctions for each are where is an arbitrary constant (not zero).In summary, the positive eigenvalues are for , and the corresponding eigenfunctions are .Python Code :1import numpy as np 2import matplotlib.pyplot as plt 3 4# Define the range of x 5x = np.linspace(-np.pi, 0, 500) 6 7# Define the eigenfunctions 8def eigenfunction(n, x): 9 mu = 0.5 + n 10 return np.cos(mu * x) 11 12# Plotting the eigenfunctions for n = 1, 2, 3, 4, 5 13plt.figure(figsize=(10, 6)) 14for n in range(1, 6): 15 y = eigenfunction(n, x) 16 plt.plot(x, y, label=f'n = {n}') 17 18plt.title('Eigenfunctions for n = 1, 2, 3, 4, 5') 19plt.xlabel('x') 20plt.ylabel('y') 21plt.legend() 22plt.grid(True) 23plt.show() 24Code Explanation :1. Import Libraries - `numpy`: Used for numerical operations. Here, it's primarily used to create an array of values for x over which the eigenfunctions will be evaluated. - `matplotlib.pyplot`: A plotting library used for creating graphs. It's used to plot and visualize the eigenfunctions.2. Define the Range of x - `x = np.linspace(-np.pi, 0, 500)`: This creates an array of 500 points linearly spaced between −π and 0. These points represent the domain over which the eigenfunctions will be plotted.3. Define the Eigenfunction - `def eigenfunction(n, x)`: This is a function definition in Python. It calculates the eigenfunction for a given n and array of x values. The eigenfunction is defined as cos⁡((12+n)x).4. Plotting the Eigenfunctions - The `for` loop (`for n in range(1, 6):`) iterates over the values of n from 1 to 5. - Inside the loop, `y = eigenfunction(n, x)` computes the eigenfunction for each n. - `plt.plot(x, y, label=f'n = {n}')`: This plots the eigenfunction for each n against x. - The `label` argument in `plt.plot` is used to create a legend for the plot.5. Customizing and Displaying the Plot - `plt.title`, `plt.xlabel`, and `plt.ylabel` are used to set the title and labels of the plot. - `plt.legend()` adds a legend to the plot. - `plt.grid(True)` adds a grid to the plot for better readability. - `plt.show()` displays the final plot.When executed, this code will produce a graph showing the different eigenfunctions for n=1,2,3,4,5 over the interval ππ[−π,0].Code Output :