Hello Bilgek. Welcome to PF. We have some very bright people in here better than me that I believe can help you. I await their comments. In the interim period may I ask the following:
For the case of f=1, I get:
[tex]2x^{3/2}z^{'}z^{''}=6x^{1/2}(z^{'})^2+4(z^{'})^{5/2}[/tex]
(just use x to make it easy to read for me)
You indicated you solved this analytically. You mind explaining how?
Also, I like the terminology of "singular initial value problem". Do you expect a unique solution? The reason is that I recently worked on:
[tex]ty^{''}+(t1)y^{'}+y=t^2;\quad y(0)=0,\quad y^{'}(0)=0[/tex]
Yea, I know it's a baby one compared to yours.
Anyway, I obtained a nonunique soltuion at t=0. I would expect the same for your equation. Are you?
