Solved Examples of Non-Homogeneous BVPs Using Eigenfunction Expansion

📚 Quick Study Guide

• 🔍 Eigenfunction Expansion: A method to solve non-homogeneous BVPs by expressing the solution as an infinite series of eigenfunctions of a related homogeneous problem.
• 💡 General Form of Non-Homogeneous BVP: $L[y(x)] = f(x)$, subject to boundary conditions, where $L$ is a linear differential operator.
• 📝 Homogeneous Problem: $L[y(x)] = 0$, with the same boundary conditions. Solve this to find the eigenfunctions $\phi_n(x)$.
• ➕ Eigenfunction Expansion Formula: $y(x) = \sum_{n=1}^{\infty} a_n \phi_n(x)$, where the coefficients $a_n$ are to be determined.
• ➗ Calculating Coefficients: $a_n = \frac{\int_a^b f(x) \phi_n(x) w(x) dx}{\int_a^b \phi_n^2(x) w(x) dx}$, where $w(x)$ is a weight function.
• 📌 Weight Function: Determined by the form of the differential operator $L$. Often, $w(x) = 1$.
• 🧮 Orthogonality: Eigenfunctions are orthogonal with respect to the weight function: $\int_a^b \phi_m(x) \phi_n(x) w(x) dx = 0$ for $m \neq n$.

Practice Quiz

1. Which of the following describes the first step in solving a non-homogeneous BVP using eigenfunction expansion?

   1. Finding the particular solution directly.
   2. Solving the related homogeneous problem.
   3. Applying Green's function method.
   4. Ignoring the boundary conditions.

2. What is the general form of the eigenfunction expansion?

   1. $y(x) = \sum_{n=1}^{\infty} a_n x^n$
   2. $y(x) = \sum_{n=1}^{\infty} a_n \phi_n(x)$
   3. $y(x) = a_0 + a_1 x$
   4. $y(x) = e^x \sum_{n=1}^{\infty} a_n$

3. How are the coefficients $a_n$ typically calculated in the eigenfunction expansion?

   1. Using Fourier transforms.
   2. Using orthogonality of eigenfunctions.
   3. By guessing a solution.
   4. Through Laplace transforms.

4. What role does the weight function $w(x)$ play in the calculation of coefficients?

   1. It normalizes the boundary conditions.
   2. It appears in the integral for calculating $a_n$.
   3. It simplifies the differential equation.
   4. It is always equal to 1.

5. What is the significance of the orthogonality of eigenfunctions?

   1. It makes the problem non-linear.
   2. It simplifies the calculation of coefficients $a_n$.
   3. It ensures the solution diverges.
   4. It has no significance.

6. Consider the BVP: $y'' + 4y = x$, $y(0) = 0$, $y(\pi/2) = 0$. What are the eigenfunctions for the related homogeneous problem?

   1. $\phi_n(x) = \sin(nx)$
   2. $\phi_n(x) = \cos(nx)$
   3. $\phi_n(x) = e^{nx}$
   4. $\phi_n(x) = x^n$

7. For a given non-homogeneous BVP, if the homogeneous solution is trivial, what does this imply about the existence of a solution to the non-homogeneous problem?

   1. The non-homogeneous problem has no solution.
   2. The non-homogeneous problem has a unique solution.
   3. The non-homogeneous problem has infinitely many solutions.
   4. The non-homogeneous problem's solution depends only on boundary conditions.