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Expert-verified Found in: Page 443 ### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069 # Question: In Problems 1–10, determine all the singular points of the given differential equation.2. x2y"-3y-xy = 0

The only singularity point exists in this differential equation for both P(x) and Q(x) is at x = 0.

See the step by step solution

## Step 1: Ordinary and Singular Points.

A point ${x}_{0}$ is called an ordinary point of equation u"+p(x)y'+q(x)y = 0 if both p and q are analytic at ${x}_{0}$ . If ${x}_{0}$ is not an ordinary point, it is called a singular point of the equation.

## Step 2: Find the singular points.

The given differential equation is

x2y"-3y'-xy = 0

Dividing the above equation by x2 we get,

$y"-\frac{3}{{x}^{2}}y\text{'}-\frac{x}{{x}^{2}}y=0$

On comparing the above equation with y"+p(x)y'+q(x)y = 0, we find that,

P(x)=$-\frac{3}{{x}^{2}}$

Q(x) =$-\frac{x}{{x}^{2}}$

=$-\frac{1}{x}$

Hence, P(x) and Q(x) are analytic except, perhaps, when their denominators are zero.

For P(x) this occurs at x = 0.

We see that P(x) is actually analytic at x = 0 as well as Q(x) is analytic except at x = 0.

The only singularity point exists in this differential equation for both P(x) and Q(x) is at x = 0. ### Want to see more solutions like these? 