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 Sci Advisor HW Helper P: 1,593 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: $$2x^{3/2}z^{'}z^{''}=6x^{1/2}(z^{'})^2+4(z^{'})^{5/2}$$ (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: $$ty^{''}+(t-1)y^{'}+y=t^2;\quad y(0)=0,\quad y^{'}(0)=0$$ Yea, I know it's a baby one compared to yours. Anyway, I obtained a non-unique soltuion at t=0. I would expect the same for your equation. Are you?