Could someone tell me the steps of solving this 2nd order differential equation?

(x^2)y'' - x y' + y = 8(x^3)

Is it Euler's equations for the left side and method of undertermined coeffficients for the right side? Thanks!!!

Question

Could someone tell me the steps of solving this 2nd order differential equation?

(x^2)y'' - x y' + y = 8(x^3)

Is it Euler's equations for the left side and method of undertermined coeffficients for the right side? Thanks!!!

anonymous · Answer

the A.E. is D^2 - D +1 =0
D= 1+-  SQRT(3) i/ 2

yc= e^1/2 (C1 cos sqrt(3)/2 + C2 sin sqrt(3)/2)

anonymous · Answer

particular solution = 1/ (D^2 - D +1) * 8(x^3)

anonymous · Answer

sry an error in typing.
yc= e^1/2 (C1 cos sqrt(3)/2 x + C2 sin sqrt(3)/2 x)

anonymous · Answer

ah again 
yc= e^1/2 x (C1 cos sqrt(3)/2 x + C2 sin sqrt(3)/2 x)

anonymous · Answer

why did you write: D^2 - D +1 =0 at the very beginning? Did you divide both sides by x squared?

anonymous · Answer

this is auxiliary equation

anonymous · Answer

yeah but you can't just ignore the x squared right? i know the quadratic formula but i am not sure if this is the case here

anonymous · Answer

still there buddy?

anonymous · Answer

ok give me a sec...

anonymous · Answer

so here we go...

anonymous · Answer

put z=logx and xD=theta , x^2D^2= theta (theta -1)

anonymous · Answer

1 sec

anonymous · Answer

Your way may work but I don't think I've seen it in class...it's gotta be easier than we think

anonymous · Answer

ok

anonymous · Answer

thanks a lot though

anonymous · Answer

you've been super helpful

anonymous · Answer

no prob... i wish i cud help u a lot more.. :(