Question

Determine a formal eigenfunction series expansion for the solution of the given problem. Assume that \(f\) satisfies the conditions of Theorem \(11.3 .1 .\) State the values of \(\mu\) for which the solution exists. $$ y^{\prime \prime}+\mu y=-f(x), \quad y(0)=0, \quad y^{\prime}(1)=0 $$

Short answer

Short answer: The formal eigenfunction series expansion for the solution to the given problem is: $$ y_P(x) = \sum_{n=1}^{\infty} a_n \sin(n \pi x), $$ where the coefficients \(a_n\) are given by: $$ a_n = \frac{-1}{\lambda_n}\int_0^1 {f(x)\sin(n \pi x) dx}, $$ and the solution exists for \(\mu = (n \pi)^2, n = 1, 2, 3, \ldots\).

Step-by-step solution

Identify the homogeneous problem

Identify the homogeneous problem associated with our given problem: $$ y'' + \mu y = 0, \quad y(0) = 0, \quad y'(1) = 0 $$ This is a homogeneous second-order ODE with boundary conditions.

Solve the homogeneous problem for eigenvalues and eigenfunctions

Solve the homogeneous problem for eigenvalues (\(\lambda\)) and corresponding eigenfunctions (\(y_n(x)\)): Depending on \(\mu\), we can have the following cases: Case 1: \(\mu > 0\), we can try the solution of the form \(y(x)=A\sin(kx)+B\cos(kx)\). The boundary conditions give us: $$ A\sin(0) + B\cos(0) = 0 \Rightarrow B=0 $$ and $$ A\cos(k)+0\sin(k)=0 \Rightarrow A\cos(k)=0 $$ From the above equation, for nontrivial solutions, \(k_n=n\pi\) and \(n=1,2,3,\ldots\). The corresponding eigenvalues for this case are \(\lambda_n = k_n^2 = (n\pi)^2\) and eigenfunctions are: $$ y_n(x) = A_n\sin(n \pi x) $$ Case 2: \(\mu < 0\), the solutions will be exponential in form, and they won't satisfy the given boundary conditions for any non-trivial solution. Therefore, we will not have any eigenvalues and eigenfunctions for this case.

Construct the eigenfunction series expansion for the particular solution

Now, we can construct the eigenfunction series expansion for the particular solution \(y_P(x)\) by superposing the eigenfunctions weighted by coefficients that are determined by the inhomogeneous term \(-f(x)\): $$ y_P(x) = \sum_{n=1}^{\infty} a_n \sin(n \pi x) $$ where \(a_n\) are the coefficients to be determined in terms of \(f(x)\).

Determine the coefficients \(a_n\)

To find the coefficients \(a_n\), multiply both sides of the problem equation by \(\sin(n \pi x)\) and integrate over the interval \((0,1)\): $$ \int_0^1 {-f(x)\sin(n \pi x) dx} = \int_0^1 {\left(y''(x) + \mu y(x)\right) \sin(n \pi x) dx} $$ Using integration by parts to rewrite the above integral, and using our eigenfunction series expansion for \(y_P(x)\), we obtain: $$ -\int_0^1 {f(x)\sin(n \pi x) dx} = a_n \lambda_n $$ The coefficients \(a_n\) can now be determined as: $$ a_n = \frac{-1}{\lambda_n}\int_0^1 {f(x)\sin(n \pi x) dx} $$

Determine the values of \(\mu\) for which the solution exists

The values of \(\mu\) we used to solve the homogeneous problem are \(\mu = (n \pi)^2\). For these values, we can always find eigenfunctions, required for the eigenfunction series expansion of our solution. Therefore, the solution exists for \(\mu = (n \pi)^2, n = 1, 2, 3, \ldots\).

Key concepts

Eigenvalues

When solving differential equations, **eigenvalues** are crucial to finding solutions that satisfy specific conditions. In the given problem, we solve for eigenvalues by analyzing the equation \(y'' + \mu y = 0\) with boundary conditions \(y(0) = 0\) and \(y'(1) = 0\). Eigenvalues are the values of \(\mu\) that allow non-trivial solutions to exist.

• For \(\mu > 0\), we assume solutions of the form \(y(x) = A\sin(kx) + B\cos(kx)\). Using boundary conditions, we find that these functions must take specific forms.
• If \(\mu < 0\), the solution becomes exponential, which does not fit the boundary conditions, leading to no eigenvalues in this case.

The eigenvalues provide critical values needed for constructing the series expansion.

Boundary Value Problems

**Boundary value problems** involve finding solutions to differential equations that satisfy specific conditions at the boundaries of the domain. For the problem \(y'' + \mu y = 0\), boundary conditions are given as \(y(0) = 0\) and \(y'(1) = 0\).

• The boundary \(y(0) = 0\) implies that the function starts at zero.
• Another condition \(y'(1) = 0\) requires that the derivative is zero at \(x = 1\).

These conditions constrain possible solutions, allowing us to determine specific eigenfunctions like \(y_n(x) = A_n\sin(n \pi x)\), used to create a series solution.

Second-Order ODE

A **second-order ordinary differential equation (ODE)** is characterized by having the second derivative in the equation. In this exercise, \(y'' + \mu y = -f(x)\) is a second-order ODE.

• The homogeneous part \(y'' + \mu y = 0\) is solved to find the fundamental nature of the solution.
• The inhomogeneous term \(-f(x)\) is treated separately and influences the coefficients in the series solution.

Second-order ODEs like this are common in physics and engineering, often describing systems like vibrations or heat distribution, where exact initial and boundary conditions need to be met.