Consider this problem:2²y" + ry/ + =0,y(1) = 0 y(2) = 0.a) Place the differential equation in standard Sturm-Liouville form, showing the steps used to do so.b) Identify and state the weighting function.c) Are the eigenvalues of this problem real?d) State the orthogonality condition for the eigenfunctions of the problem.You do not need to solve for the eigenfunctions, nor do you need to state the eigenvalues.

The differential equation given here is x2y''+xy'+λx=0 y''+y'x+λx3=0 compare it with equation…

Answered: Consider this problem: r²y" + xy/ + ? =…

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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Consider this problem:
2²y" + ry/ + =
0,
y(1) = 0 y(2) = 0.
a) Place the differential equation in standard Sturm-Liouville form, showing the steps used to do so.
b) Identify and state the weighting function.
c) Are the eigenvalues of this problem real?
d) State the orthogonality condition for the eigenfunctions of the problem.
You do not need to solve for the eigenfunctions, nor do you need to state the eigenvalues.

Expert Solution

Step 1

The differential equation given here is

$x^{2} y^{''} + x y^{'} + \frac{λ}{x} = 0$

$y^{''} + \frac{y^{'}}{x} + \frac{λ}{x^{3}} = 0$

compare it with equation

$y'' + y^{'} p (x) + q (x) y = 0$

$\frac{P^{'} (x)}{P (x)} = p (x) \Rightarrow P (x) = c e^{\int p (x) d x}$

$P (x) = c e^{\int \frac{1}{x}} d x = c x$

let c= 1 and $λ w - Q = P (x) q (x)$

$λ w - Q = c x λ / x^{3}$

So Q = 0 and $w = 1 / x^{2}$

So, Sturm - Liouville form is given as

$- \frac{\partial}{\partial x} (p (x) \frac{\partial y}{\partial x}) + q (y) y = w λ y$

plugging the values,

$- \frac{\partial}{\partial x} (c x \frac{\partial y}{\partial x}) + \frac{λ}{x^{3}} y = \frac{1}{x^{2}} λ y$

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Consider this problem: r²y" + xy/ + ? = - 0, y(1) = 0 y(2) = 0. a) Place the differential equation in standard Sturm-Liouville form, showing the steps used to do so.