Okay, you're up to this point :
Now divide by v (if v=0, it's a nonsense) :u''v+u'(2v'+pv)+u(v''+pv'+qv)=0.
So since in the final expression, there is no more u', solve for v in
then substitute it in and it'll give you g =)
the reason for the ? in the title is because I'm not really sure what method to use here:
Determine v(x) such that the transformation y=u(x)v(x) will transform y''+p(x)y'+q(x)y=0 into u''+g(x)u=0 where g(x) can be expressed in terms of p(x) and q(x). Your answer should express both v(x0 and g(x) in terms of p(x) and q(x).
I have tried the following:
Let y=uv (callign the functions u and v for simplicity's sake)
Then y'=uv' + vu'
y'' = uv''+2u'v'=u''v
Then I plugged these back into the original DE, and after grouping got:
I have no idea where to go from here though. any thoughts are greatly appreciated. Thanks so much!