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Q 4E

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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.4. (x2+x)y"+3y'-6xy = 0

The singular point exists in this differential equation for both P(x) and Q(x) is at x = 0,-1

See the step by step solution

## Step 1: Ordinary and Singular Points

A point ${x}_{0}$ is called an ordinary point of equation y"+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,

(x2+x)y"+3y-6xy = 0

Dividing the above equation by (x2+x)we get,

y"+$\frac{3}{\left({x}^{2}+x\right)}y\text{'}-\frac{6x}{\left({x}^{2}+x\right)}y=0$

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

P(x) = $\frac{3}{x\left(x+1\right)}$

Q(x) =$-\frac{6x}{x\left(x+1\right)}$

=$-\frac{6}{\left(x+1\right)}$..................................(1)

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

For P(x) this occurs at x = 0,-1 .

We see that Q(x) is actually analytic at x = 0 .

Since we can cancel an x in the numerator and denominator of Q(x) as shown in (1).

Therefore, P(x) is analytic except at x = 0,-1 as well as Q(x) is analytic except at x = -1 .

The singular point exists in this differential equation for both P(x) and Q(x) is at x = 0,-1 .