Answered: Use the method if eigenvalue expansions…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use the method if eigenvalue expansions to solve

y"+lamda*y=x^2

y'(0)=0, y'(1)=0

Expert Solution

Step 1

We will be solving $y^{''} + λ y = x^{2},$ along with the boundary conditions $y^{'} (0) = 0, y^{'} (1) = 0$ by the eigenvalue expansions method.

First, we will solve the corresponding homogeneous equation, that is $y^{''} + λ y = 0$ satisfying the boundary conditions $y^{'} (0) = 0, y^{'} (1) = 0$ .

Now, the auxiliary equation is $m^{2} + λ = 0$ . Then $m = \pm i \sqrt{λ}$ .

So, the general solution is $y = c_{1} \cos \sqrt{λ} x + c_{2} \sin \sqrt{λ} x$ .

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