Question

Find a second solution y2 for x^2*y"-2xy'+(x^2+2)y=0; y1=x*cos(x) that isn't a constant multiple of the solution y1.

Answer 1

\[y _{2}=ux \cos x\]

Answer 2

\[y _{2}'=u \cos x-ux \sin x+u'x \cos x\]

Answer 3

\[y _{2}''=-2u \sin x+2u'\cos x-2u'x \sin x-ux \cos x\]

Answer 4

Subbing into ODE gives...

Answer 5

\[-2x ^{3}u'\sin x=0\]

Answer 6

Now I'm stucked. @SithsAndGiggles

Answer 7

I notice in your second derivative you have no term with factor u''

Answer 8

isnt there a wronskian involved somehow?

Answer 9

I see the mistake now. Thanks for pointing out. No wonder it's not working!