Find the general solution to the differential equation x(x − 2)y′′ − (x^2 − 2)y′ + 2(x − 1)y = 0, y1(x)=e^x.

We have to use variation of parameters but the question is in such a weird form that i dont know what to do...

Help appreciated

With only one solution, this is a reduction of order question.

Let y = ue^x. Differentiating,
y' = (u' + u)e^x, and y'' = (u'' + 2u' + u)e^x.

Substituting yields
x(x - 2) * (u'' + 2u' + u)e^x - (x^2 - 2) * (u' + u)e^x + 2(x - 1) * ue^x = 0
==> (x^2 - 2x) (u'' + 2u' + u) - (x^2 - 2) (u' + u) + (2x - 2)u = 0
==> (x^2 - 2x)u'' + (x^2 - 4x + 2)u' = 0

This is linear in u'; separating variables yields
u''/u' = -(x^2 - 4x + 2)/(x^2 - 2x)
.......= -1 + (2x - 2)/(x^2 - 2x)

Integrate both sides:
ln u' = -x + ln(x^2 - 2x) + A
==> u' = B(x^2 - 2x) e^(-x), where B = e^A.

Integrate both sides again:
u = B [-(x^2 - 2x)e^(-x) - (2x - 2)e^(-x) - 2e^(-x)] + C
==> ue^x = -B [(x^2 - 2x) + (2x - 2) + 2] + Ce^x
==> y = C₁ x^2 + C₂ e^x.

I hope this helps!

I'm not sure how I would use Variation of Parameters, since this is usually used to find particular solution of non-homogeneous equation when you have a set {y₁, y₂} of fundamental solutions to the associated homogeneous equation. But here, we don't have y₂, and we don't have a non-homogeneous equation.
http://www.sosmath.com/diffeq/second/var…

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Perhaps they mean for you to use Reduction of Order ?
http://www.sosmath.com/diffeq/second/red…

We have to rewrite differential equation in the form: y'' + p(x) y' + q(x) y = 0
x(x − 2) y'' − (x² − 2) y' + 2(x − 1) y = 0
y'' − (x² − 2)/(x(x − 2)) y' + 2(x − 1)/(x(x − 2)) y = 0

Second solution y₂ (independent of y₁) can be found as:
y₂(x) = y₁(x) v(x), where