Boundary-Value Problems and Eigenvalues

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

The idea

An initial-value problem specifies a state at one point. A boundary-value problem constrains the solution at different points. Even a linear equation may then have no solution or multiple solutions. Special parameter values can permit nonzero solutions of a homogeneous boundary problem.

Visual guide

VISUAL GUIDEFixed endpoints select discrete mode shapes

n = 1
n = 2
n = 3

On [0, π], sin x, sin 2x and sin 3x all vanish at both endpoints. They correspond to eigenvalues 1, 4 and 9. Higher modes have more interior nodes; arbitrary wavelengths would fail the boundary conditions.

Method and assumptions

For $y''+\lambda y=0$ on $[0,L]$ with $y(0)=y(L)=0$ , inspect positive, zero and negative $\lambda$ separately. Nontrivial solutions exist precisely at $\lambda_n=(n\pi/L)^2$ , with eigenfunctions $\sin(n\pi x/L)$ for positive integers $n$ . The zero function always satisfies the homogeneous problem.

Worked example: a fixed-end mode

On $[0,\pi]$ , $\lambda=4$ gives $y=A\sin2x$ . Both endpoints vanish for every amplitude $A$ . The eigenfunction has one interior zero at $x=\pi/2$ , unlike the fundamental mode $\sin x$ .

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

For y″+λy=0, y(0)=y(π)=0, which positive λ admits a nonzero solution?

λ=2

λ=1

λ=1/4

Hint 1 · Find a starting point

The first condition removes the cosine term.

Hint 2 · Take the next step

The second needs sin(√λπ)=0.

Show the reasoning

Answer: λ=1

√λ must be a positive integer. λ=1 gives y=C sin x, a nontrivial family when C≠0.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Worked example: incompatible endpoints

For $y''+y=0$ on $[0,\pi]$ with $y(0)=0,y(\pi)=1$ , the first condition forces $y=A\sin x$ . But $y(\pi)=0$ for every $A$ , so no solution can meet the second condition.

Interpreting the result

In engineering, eigenfunctions describe spatial mode shapes; time evolution enters through an associated dynamic model. Orthogonality of distinct sine modes helps decompose a shape or forcing. Do not confuse the free amplitude of an eigenfunction with a unique boundary-value solution.

Practice

Give the first eigenvalue for $L=2$ .
Can $\lambda=0$ give a nonzero solution with both endpoints zero?
What is the second eigenfunction on $[0,1]$ ?

Show worked solutions

$\lambda_1=\pi^2/4$ .
No. Then $y=ax+b$ , and both boundary conditions force $a=b=0$ .
$\sin2\pi x$ , up to a nonzero constant multiple.

Further study

MIT OpenCourseWare: Differential Equations provides a full university course with additional lectures and exercises.

MAKE IT YOURS

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