how to use reduction of order to solve different equ.?

Question

How do you use the method of reduction of order to find the general solution with the given function?

xy" - 2(x+1)y' + (x+2)y = 0, Y1=e^(x)

Answer 1

Assume that y2(x) = v(x)y1(x) = v(x)e^x.

Then y2' = v'e^x + ve^x, and
y2'' = v''e^x + 2v'e^x + ve^x.

Plug these in to the equation:
x(v'' + 2v' + v)e^x - 2(x+1)(v'+v)e^x + (x+2)ve^x = 0,
or
xv'' + (2x - 2(x+1))v' + (x-2(x+1)+x+2)v = 0
Simplifying:
xv'' + v' = 0.
Let u=v'. Then xu' + u = 0, which can easily be solved for u. Then differentiate it and multiply it by y1(x) to get y2(x).