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Q - 8E

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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.8. exy"-(x2-1)y'+2xy=0

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

See the step by step solution

Step 1: Ordinary and Singular Points

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.

Step 2: Find the singular points

The given differential equation is

exy"-(x2-1)y'+2xy=0

Dividing the above equation by ex we get,

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

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

For Px) this occurs at no point. In this case both Px) and Q(x) are analytic at all points.

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

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