1. ## Differential Equation...transformation?

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:
u''v+u'(2v'+pv)+u(v''+pv'+qv)=0.

I have no idea where to go from here though. any thoughts are greatly appreciated. Thanks so much!

2. Hello,

Okay, you're up to this point :
u''v+u'(2v'+pv)+u(v''+pv'+qv)=0.
Now divide by v (if v=0, it's a nonsense) :

$u''+u' \cdot \frac{2v'+pv}{v}+u(\frac{v''}{v}+p \frac{v'}{v}+q)=0$

So since in the final expression, there is no more u', solve for v in $2v'+pv=0$

then substitute it in $\frac{v''}{v}+p \frac{v'}{v}+q$ and it'll give you g =)

3. Ah! Thank you very much! Rest of the process was easy enough...you gave me the missing pieces. Thanks again!