Problem 14 Find an eigenfunction expansion ... [FREE SOLUTION]

Chapter 17: Problem 14

Find an eigenfunction expansion for the solution with boundary conditions $y(0)=y(\pi)=0$ of the inhomogeneous equation $$ \frac{d^{2} y}{d x^{2}}+\kappa y=f(x) $$ where $\kappa$ is a constant and $$ f(x)= \begin{cases}x, & 0 \leq x \leq \pi / 2 \\ \pi-x, & \pi / 2

Short Answer

Expert verified

The eigenfunction expansion is $ y(x) = \sum_{n=1}^\infty c_n \sin(nx) $ where $ c_n = \frac{b_n}{n^2 - \kappa} $.

Step by step solution

Identify the boundary value problem

The given boundary value problem is $ \frac{d^2 y}{d x^2} + \kappa y = f(x) $ with boundary conditions $ y(0) = y(\pi) = 0 $.

Determine the corresponding homogeneous problem

Solve the homogeneous equation $ \frac{d^2 y}{d x^2} + \kappa y = 0 $ with the same boundary conditions $ y(0) = y(\pi) = 0 $.

Solve the homogeneous problem

Assuming $ y(x) = e^{\lambda x} $, the characteristic equation becomes $ \lambda^2 + \kappa = 0 $, giving us $ \lambda = \pm i\sqrt{\kappa} $. Thus, the general solution is $ y_h(x) = A\cos(\sqrt{\kappa}x) + B\sin(\sqrt{\kappa}x) $. Using the boundary conditions, solve for A and B. Applying $ y(0) = 0 $, we get A = 0. Applying $ y(\pi) = 0 $, we find $ B\sin(\sqrt{\kappa}\pi) = 0 $, leading us to $ \sqrt{\kappa} \pi = n\pi $ (where n is an integer), so $ \sqrt{\kappa} = n $. The homogeneous solution satisfying the boundary conditions is $ y_h(x) = B\sin(nx) $.

Find the particular solution

The particular solution for the inhomogeneous equation can be expressed as a series expansion $ y_p(x) = \sum_{n=1}^\infty c_n \sin(nx) $. Use Fourier sine series to find the coefficients $ c_n $.

Compute Fourier sine series coefficients

Given $ f(x) $, find the Fourier sine series coefficients $ c_n $ for $ f(x) $. The expansion is given by: \[ f(x) = \sum_{n=1}^{\infty} b_n \sin(nx) \], where \[ b_n = \frac{2}{\pi} \left( \int_0^{\pi/2} x \sin(nx) \, dx + \int_{\pi/2}^{\pi} (\pi - x) \sin(nx) \, dx \right) \]. Compute these integrals.

Solve the integrals for coefficients

Compute the integrals: \[ b_n = \frac{2}{\pi} \left( \int_0^{\pi/2} x \sin(nx) \, dx + \int_{\pi/2}^{\pi} (\pi - x) \sin(nx) \, dx \right) \]. Calculate each integral separately: \[ \int_0^{\pi/2} x \sin(nx) \, dx \] and \[ \int_{\pi/2}^{\pi} (\pi - x) \sin(nx) \, dx \]. Use integration by parts to solve these integrals.

Combine solutions

Combine the coefficients to form the series for the particular solution $ y_p(x) = \sum_{n=1}^{\infty} c_n \sin(nx)$ and add it to the homogeneous solution to form the eigenfunction expansion of the original equation. So, the final solution is $ y(x) = \sum_{n=1}^{\infty} \frac{b_n}{n^2 - \kappa} \sin(nx)$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inhomogeneous differential equations

Inhomogeneous differential equations are a type of differential equation that includes a non-zero forcing function, often denoted as $f(x)$. Unlike homogeneous differential equations, which have zero on the right-hand side, inhomogeneous equations include an additional term. In this problem, we have: \begin{itemize}

An inhomogeneous differential equation: $ \frac{d^2 y}{d x^2} + \kappa y = f(x) $ with $\kappa$ as a constant.

The forcing function $f(x)$ varies with different segments defined piecewise.

The approach to solving such equations involves:

Solving the corresponding homogeneous equation (without $f(x)$).
Finding a particular solution for the inhomogeneous equation.
Combining these solutions to get the general solution.

boundary value problems

Boundary value problems (BVPs) are differential equations accompanied by a set of additional constraints called boundary conditions. They include:

Conditions at the boundaries (endpoints) of the domain of the solution.
For our problem: $y(0)=y(\pi)=0$.

Solving BVPs often involves:

Firstly solving the homogeneous problem which satisfies the boundary conditions.
Secondly adding the particular solution of the inhomogeneous equation, which also must satisfy the boundary conditions.

This ensures that the solution fits the physical or geometric constraints posed by the boundaries.

Fourier sine series

The Fourier sine series is a way to represent a function as a sum of sine terms. It is particularly useful in solving boundary value problems, especially with conditions like $ y(0) = y(\pi) = 0 $. For our problem:

We express the given function $f(x)$ as a series of sines.
The coefficients $b_n$ of the sine series are computed using the formula: $ b_n = \frac{2}{\pi} \left( \int_0^{\pi/2} x \sin(nx) \, dx + \int_{\pi/2}^{\pi} (\pi - x) \sin(nx) \, dx \right) $.

The Fourier sine series coefficients help us find the particular solution to the differential equation. After calculating these coefficients, the particular solution is summed as an infinite series of sine terms.

homogeneous solutions

Homogeneous solutions are solutions to the differential equation without the non-homogeneous term ($f(x)$). For our problem, we solve:

The equation: $\frac{d^2 y} {d x^2} + \kappa y = 0 $.
By assuming a solution form like $y(x) = e^{\lambda x}$, we solve the characteristic equation $\lambda^2 + \kappa = 0$.

This gives us:

A general solution $y_h(x) = A \cos(\sqrt{\kappa} x) + B \sin(\sqrt{\kappa} x)$.
Using the boundary conditions $ y(0) = 0 $ and $ y(\pi) = 0 $, we determine coefficients $A$ and $B$.

For this problem:

We find $ A = 0 $ and $ \sqrt{\kappa} = n $
Thus homogeneous solution becomes $ y_h(x) = B \sin(nx) $.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

91影视

Find an eigenfunction expansion for the solution with boundary conditions \(y(0)=y(\pi)=0\) of the inhomogeneous equation $$ \frac{d^{2} y}{d x^{2}}+\kappa y=f(x) $$ where \(\kappa\) is a constant and $$ f(x)= \begin{cases}x, & 0 \leq x \leq \pi / 2 \\ \pi-x, & \pi / 2