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Q 4E

Expert-verifiedFound in: Page 443

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. (x ^{2}+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

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.

The given differential equation is,

(x^{2}+x)y"+3y-6xy = 0

Dividing the above equation by (x^{2}+x)we get,

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

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

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

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

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

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