Answer to Question #285268 in Differential Equations for Pavan

Question

F or the follow ing differential equation locate and classify its singular points on the x-axis

𝑥

2

𝑦

′′ + (2 − 𝑥 )𝑦

′ = 0.

Answer 1

Solution:

𝑥²𝑦′′+ (2−𝑥)𝑦′= 0.

$y''+\frac{2-x}{x^2}y'=0$

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

x = 0 is singular point

$(x-0)P(x)=\frac{2-x}{x}$ is not analytic at x = 0

so, x = 0 is irregular singular point

x = 2 is singular point

$(x-2)P(x)=-\frac{(x-2)^2}{x^2}$ ​is analytic at x = 2

so, x = 2 is regular singular point