Question

ODE:
Using the fact that Po(x) = 1 is a solution of
(1-x^2)y'' - 2xy' = 0,
find a second independent solution using the method of reduction of order of a homogeneous equation.

Answer 1

Let v = y'. Hence v' = y''.

So (1-x^2)v' - 2xv = 0
2xv = (1-x^2)(dv/dx)
2x / (1-x^2) dx = (1/v) dv
-ln (1-x^2) = ln v + K
ln (1-x^2)^-1 = ln (Cv)
Cv = 1 / (1-x^2)
y' = A / (1-x^2)
y' = X / (1-x) + Y / (1+x)

Using partial fractions, X = A / 2 and Y = -A / 2

Then integrate y' to get y.