Question

Solve the given problem by means of an eigenfunction expansion. $$ y^{\prime \prime}+2 y=-x, \quad y(0)=0, \quad y(1)=0 $$

Short answer

Question: Find the solution for the given non-homogeneous differential equation with boundary conditions: $$y^{\prime\prime} + 2y = -x, \quad y(0)=0, \quad y(1)=0$$ Answer: The solution to the given non-homogeneous differential equation is: $$y(x) = \sum_{n=1}^{\infty} \frac{(-1)^n - 1}{(n\pi)^2} \sin(n\pi x)$$

Step-by-step solution

Find the eigenfunctions and eigenvalues

To find the eigenfunctions and eigenvalues, let's consider the homogeneous equation associated with the given problem: $$ y^{\prime\prime} + 2y = 0 \quad \text{with} \quad y(0)=0, \quad y(1)=0 $$ Let's assume the eigenvalues to be of the form \(\lambda = -k^2\). Then the homogeneous equation becomes: $$ y^{\prime\prime} + 2k^2 y = 0 $$ The general solution of this differential equation is: $$ y(x) = C_1 \cos(kx) + C_2 \sin(kx) $$ Now, applying the boundary conditions, we get: $$ y(0) = C_1 \cos(0) + C_2 \sin(0) \Rightarrow C_1 = 0 $$ and $$ y(1) = C_1 \cos(k) + C_2 \sin(k) \Rightarrow C_2 \sin(k) = 0 $$ To have a non-trivial solution, we need \(\sin(k) \neq 0\), which gives us the eigenvalues \(k = n\pi\), where \(n = 1, 2, 3, ...\) is an integer. The corresponding eigenfunctions are: $$ y_n(x) = \sin(n\pi x) $$

Expand the given function in terms of eigenfunctions

Now, let's expand the given function (-x) in terms of the eigenfunctions. To do this, we need to find the coefficients \(b_n\) for the Fourier sine series: $$ -x = \sum_{n=1}^{\infty} b_n \sin(n\pi x) $$ Calculating the Fourier coefficients, we find: $$ b_n = 2 \int_0^1 (-x) \sin(n\pi x) \, dx $$ Solving the integral, we get: $$ b_n = \frac{2}{n\pi}[(-1)^n - 1] $$

Use the Fourier series expansion method

Now, the solution for the given non-homogeneous differential equation can be expressed as: $$ y(x) = \sum_{n=1}^{\infty} c_n y_n(x) $$ where \(c_n\) represents the coefficients and \(y_n(x)\) are the eigenfunctions. Using the orthogonality of the eigenfunctions, we have: $$ c_n = \frac{b_n}{2n\pi} $$ Thus, the final solution to the given problem is: $$ y(x) = \sum_{n=1}^{\infty} \frac{(-1)^n - 1}{(n\pi)^2} \sin(n\pi x) $$

Key concepts

Differential Equation

A differential equation is a mathematical equation that involves the derivatives of a function. It describes how a function changes and is an essential tool in modeling the real world phenomena where change is involved. In this exercise, we deal with a second-order differential equation:\[ y'' + 2y = -x \]The primes denote derivatives, so \(y''\) refers to the second derivative of \(y\) with respect to \(x\). This equation is not only about finding \(y\) but also understanding how \(y\) changes as \(x\) changes. The term \(-x\) on the right side makes this a non-homogeneous equation, which means it has an external input or force affecting it.

Boundary Value Problem

Boundary value problems involve differential equations that need to satisfy specific conditions at multiple points, known as the boundaries. These problems are essential in physics and engineering where solutions are not only expected to be continuous but also to meet specific constraints. In our exercise, the boundary conditions are:

• \(y(0) = 0\)
• \(y(1) = 0\)

These constraints mean the solution \(y\) must be zero at both \(x = 0\) and \(x = 1\). Solving boundary value problems often involves techniques like eigenfunction expansion, which help in finding solutions that adhere to these constraints.

Fourier Series

The Fourier series is a method to express a function as the sum of sine and cosine terms. It is an excellent tool for breaking down complex periodic problems into simpler parts that are easier to analyze. In this exercise, the function \(-x\) is expanded using a Fourier sine series in terms of sine functions because of the given boundary conditions at 0 and 1. The expression:\[-x = \sum_{n=1}^{\infty} b_n \sin(n\pi x)\]uses coefficients \(b_n\), found by integrating the product of the function and the sine function over the range [0,1]. Essentially, a Fourier series allows the problem to be transformed into a series of independent simpler problems.

Eigenvalues and Eigenfunctions

Eigenvalues and eigenfunctions are fundamental concepts in solving differential equations, especially with boundary conditions. An eigenfunction satisfies a given differential equation for corresponding eigenvalues. In this exercise, we determine the eigenvalues by imposing conditions on the solution:\[ y_n(x) = \sin(n\pi x) \]where \(n\) is an integer. Eigenvalues \(k = n\pi\) arise from the requirement that the system needs to be stable with non-trivial solutions. Eigenfunctions \(\sin(n\pi x)\) form a basis that helps in representing the solution through a series. They ensure that the solution meets all specified boundary conditions while allowing the equation to be solved analytically.