Solving D.E. with Variation of Parameters Technique

[RadiationX]

I need to solve this D.E.

$$x^2y''-xy' + y = x^3$$

i'm supposed to use the variation of parameters technique.

in that technique i need to get a coeffecient of 1 in the first postion of y'' and then sove the homogenous D.E.

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

the above leads to

$$m^2-2m +1=0$$

now solving this i get

$$C_1x +C_2tx$$

my problem is that i don't know how to move forward with
$$C_2tx$$

how do i proceed

 

[dextercioby]

It's an Euler equation.You need to transform it into a linear equation with constant coeffs.Read the theory again and identify the substitution you need.

Daniel.

 

[RadiationX]

do you mean that this is a Bernoulli equation?

 

[dextercioby]

I told u it was/is an Euler eqn.

Make the sub

$$x=e^{t}$$

$$y(x)\longrightarrow \bar{y}(t)$$

Daniel.

 

[RadiationX]

ok. i'll figure it out