Question

Find a formal solution of the nonhomogencous boundary value problem $$ -\left(x y^{\prime}\right)^{\prime}=\mu x y+f(x) $$ \(y, y^{\prime}\) bounded as \(x \rightarrow 0, \quad y(1)=0\) where \(f\) is a given continuous function on \(0 \leq x \leq 1,\) and \(\mu\) is not an eigenvalue of the corresponding homogeneous problem. Hint: Use a series expansion similar to those in Section \(11.3 .\)

Short answer

Answer: The coefficients \(a_n\) are found by multiplying both sides of the equation by \(\phi_m(x)\), integrating from 0 to 1, and using the orthogonality relationship. The coefficients are given by the formula \(a_m = \frac{\int_0^1 f(x) \phi_m(x) dx}{\int_0^1 \mu x \phi_m^2(x) dx}\), for \(m=1, 2, \dots\).

Step-by-step solution

Set up the homogeneous problem

First, let's rewrite the given nonhomogeneous boundary value problem into its homogeneous counterpart: $$ -\left(x y_H^{\prime}\right)^{\prime}=\mu x y_H, $$ where \(y_H(x)\) is the solution of the homogeneous problem, with boundary conditions: $$ y_H, y_H^{\prime}\text{ bounded as } x \rightarrow 0, \quad y_H(1)=0. $$

Solve the homogeneous problem

We start by solving the homogeneous problem by looking for eigenvalues and eigenfunctions. To do this, consider the equation: $$ -\left(x y_H^{\prime}\right)^{\prime}=\mu x y_H. $$ This is an ordinary differential equation (ODE), which looks like a Sturm-Liouville problem. The general solution format would be: $$ y_H(x) = \sum_{n=1}^{\infty} a_n \phi_n(x), $$ where \(\phi_n(x)\) are the eigenfunctions for the corresponding eigenvalues of the homogeneous problem. To find the eigenfunctions, we need to solve the ODE: $$ -\left(x y_H^{\prime}\right)^{\prime}=\lambda x y_H. $$ This equation needs to be solved subject to the boundary conditions from the problem above. Solving the ODE, we find the eigenfunctions.

Find eigenvalues and eigenfunctions

Solve the ODE to obtain eigenvalues and eigenfunctions. Let's denote eigenvalues as \(\lambda_n\) and eigenfunctions as \(\phi_n(x)\). After solving the ODE, you will obtain a set of eigenvalues and corresponding eigenfunctions.

Expand the solution of nonhomogeneous problem

Now we use the eigenfunctions found in step 3 to expand the solution of the nonhomogeneous problem based on the hint. We can express the solution \(y(x)\) as: $$ y(x) = \sum_{n=1}^\infty a_n \phi_n(x), $$ where \(a_n\) are coefficients to be determined.

Determine the coefficients \(a_n\)

Substitute the expression for \(y(x)\) back into the nonhomogeneous equation: $$ -\left(x \left(\sum_{n=1}^\infty a_n \phi_n^{\prime}(x)\right)^{\prime}\right) = \mu x \sum_{n=1}^\infty a_n \phi_n(x) + f(x). $$ Now, we need to find the coefficients \(a_n\) that satisfy this equation. To find \(a_n\), we multiply both sides of the equation by \(\phi_m(x)\), integrate from 0 to 1, and use the orthogonality relationship: $$ a_m = \frac{\int_0^1 f(x) \phi_m(x) dx}{\int_0^1 \mu x \phi_m^2(x) dx}, $$ for \(m=1, 2, \dots\). We can find each coefficient \(a_m\) by evaluating these integrals.

Write down the formal solution

With the coefficients \(a_n\) determined, we can write down the formal solution of the nonhomogeneous problem as: $$ y(x) = \sum_{n=1}^\infty a_n \phi_n(x), $$ where the \(a_n\) coefficients are given by the formula in Step 5 and the \(\phi_n(x)\) are the eigenfunctions obtained from solving the homogeneous problem in Steps 2 and 3.

Key concepts

Series Expansion

In solving boundary value problems, series expansion is a powerful method to express complex functions as sums of simpler, more manageable components. Essentially, this involves breaking down a function into a series of terms that are easier to handle analytically.
For our problem, we consider the solution of both the homogeneous and nonhomogeneous boundary value problem as a series. This is expressed in the general form:

• \[ y(x) = \sum_{n=1}^\infty a_n \phi_n(x) \]

The terms \( \phi_n(x) \) represent the eigenfunctions, which we will derive from solving the associated homogeneous problem.
The coefficients \( a_n \) are constants that we need to determine based on specific conditions of the problem.
Breaking down the solution into series form allows us to leverage the power of orthogonality and other properties, simplifying the often challenging task of determining the explicit solution to differential equations.

Sturm-Liouville Problem

The concept of a Sturm-Liouville problem is central to understanding complex differential equations in boundary value problems. It is a specific type of self-adjoint linear differential equation, defined as:

• \[ -(p(x) y')' + q(x) y = \lambda w(x) y \]

where \( p(x), q(x), \) and \( w(x) \) are given functions, and \( \lambda \) are the eigenvalues.
This structure ensures certain properties, such as orthogonality, which become very useful in finding solutions.
The problem given involves rearranging our boundary value problem into a form that exhibits Sturm-Liouville characteristics.
We introduce the homogeneous counterpart:

• \[ -(x y_H')' = \mu x y_H \]

Transforming the boundary value equation in this manner allows us to use well-established methods to find its eigenvalues and eigenfunctions.

Eigenvalues and Eigenfunctions

Eigenvalues and eigenfunctions are crucial in solving differential equations, especially when approaching a boundary value problem in the Sturm-Liouville form. They provide the foundational elements to express a solution in a series expansion.
Here, we solve

• \[ -(x y_H')' = \lambda x y_H \]

to obtain the values of \( \lambda_n \) (eigenvalues).
Each eigenvalue corresponds to an eigenfunction \( \phi_n(x) \), which describes a mode of behavior within the system being analyzed. These functions form an orthogonal set on the given interval.
Understanding the eigenvalues and eigenfunctions means we can integrate them to construct solutions to the differential equation.
This method greatly simplifies the task of finding the coefficients used in series solution expansions.

Orthogonality Relationship

Orthogonality in the context of eigenfunctions is an essential property that allows for the simplification of the system of equations obtained from a differential equation.
When eigenfunctions are orthogonal, this means:

• \[ \int_0^1 \phi_n(x) \phi_m(x) \, dx = 0 \quad \text{for } n eq m \]

This property simplifies the calculation of series coefficients \( a_n \).
We exploit orthogonality in finding the coefficients by integrating across the interval with respect to \[ \phi_m(x) \]. This yields a solution for \( a_m \) as shown:

• \[ a_m = \frac{\int_0^1 f(x) \phi_m(x) \, dx}{\int_0^1 \mu x \phi_m^2(x) \, dx} \]

Utilizing orthogonal properties makes it significantly easier to isolate and calculate individual coefficients, as overlap between different eigenfunctions does not contribute to the solution.