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Q - 8E

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.**

**8. e ^{x}y"-(x^{2}-1)y'+2xy=0**

There is no singularity point that exists in this differential equation for both ^{P(X)} and Q(X).

A point ^{X0} is called an ordinary point of equation y"+p(x)y'+q(x)y=0 if both p* *and q* *are analytic at ^{X0} . If ^{X0} is not an ordinary point, it is called a singular point of the equation.

The given differential equation is

e^{x}y"-(x^{2}-1)y'+2xy=0

Dividing the above equation by e^{x} we get,

$y"-\frac{({x}^{2}-1)}{{e}^{x}}y\text{\'}+\frac{2x}{{e}^{x}}y=0$

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

$Q\left(x\right)=\frac{2x}{{e}^{x}}$

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

For ^{P}^{x)} this occurs at no point. In this case both ^{P}^{x)} and^{ Q(x)} are analytic at all points.

Therefore, there is no singularity point that exists in this differential equation for both ^{P}^{x)} and^{ Q(x)} .

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