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I have the 2nd-order nonlinear ODE below:

[tex]

k(v)=\frac{\phi ''(v)}{\phi (v) (\phi ' ^2 (v) +1)^2}

[/tex]

Where k(v) is some function. I would like to investigate for what functions k there can exist solutions on a given interval [tex][a,b][/tex]. For example, if [tex]k(v)=0[/tex], then [tex]\phi '' (v)=0[/tex] which implies that [tex]\phi (v)=C_1 v + C_2[/tex]. The DE gets very complicated very quickly, though, and I'm not sure how to approach the problem.

I do not know very much about differential equations, so I need help in figuring out what to learn. What are the methods for analyzing such a DE? Are there any existence theorems or uniqueness theorems already out there? Can anyone recommend some good literature?

Thanks for the help.

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# 2nd-Order Non-Linear ODE: Where to Start?

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