# ordinary differential equation: $(f'(z))^2 = c\,f(z)^3 + f(z)^2$

I swear I have seen this type of ODE before, but I can't remember how to attack it. In general, I would like to know how to solve $$\left(f'(z)\right)^m = c\,G(z)^n$$ where $m,\;n \in \mathbb{N}$ and $G(z)$ is just a polynomial in $f(z)$.

This sounds hard, so I would be happy with, $$\left(f'(z)\right)^2 = c\,G(z)^n,$$ though this latter equation may be too difficult too. For my homework, though, I need to know, $$\left(f'(z)\right)^2 = \left( c\,f^3 + f^2 \right).$$ It was also suggested in the homework question to utilize $$g^2 = 3\,c-f.$$ If I do this without thinking I get an equation $$\left(f'(z)\right)^2 = c\,f(z)^2,$$ which seems much easier, though I am still a little rattled by the plus-minus.

In case it matters, this is a related to a method for solving the Korteweg-deVries equation, $$u_t + u\,u_x + u_{xxx} = 0.$$ I have seen some solutions (but did not understand them) where the polynomial was "factored" into 3 roots... something like that. I just don't want to know the answer, but how to get it. Please keep in mind that this is my first class in PDEs.

Thanks for any help!

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Yes, I was there too.... I was thinking for some reason there may be some complications due to the sign of the square root. Anyway, thanks! –  nate Feb 1 '12 at 0:30
"I have seen some solutions (but did not understand them) where the polynomial was "factored" into 3 roots" - and that is precisely the Weierstrass approach. –  Ｊ. Ｍ. Feb 8 '12 at 18:17

\begin{align*} \left(\frac{dy}{dz}\right)^2 &= cy^3+y^2\\ \frac{dy}{y \sqrt{cy+1}} &= dz\\ cy+1 &= u\\ 2 \frac{du}{u^2-1} &= dz\\ - 2 \tanh^{-1} u + C&= z\\ u &= \tanh \frac{-z+C}{2}\\ \sqrt{cy+1} &= -\tanh \frac{z-C}{2}\\ cy+1 &= \tanh^2 \frac{z-C}{2}\\ y &= \frac{\tanh^2 \frac{z-C}{2}-1}{c}\\ y &= -\left({c\times \cosh^2 \frac{z-C}{2}}\right)^{-1} \end{align*}
Aaahhh, I think that the $g^2 = 3\,c-f$$must have been a hint towards a substitution in the integration. Turns out it was just confusing the way it was suggested in the problem. Now I see it as a straight-forward problem. Thanks all! – nate Feb 1 '12 at 1:16 There is this famous differential equation ...$$ [\wp'(z)]^2 = 4[\wp(z)]^3 - g_2\wp(z) - g_3 $$Weierstrass Elliptic Functions - Ahh. Thank you - I was also wondering how I was going to solve this pde if I wasn't able to get rid of 2 BC's - would lead to a third-order polynomial in p(z) just like you mentioned. Thanks! – nate Feb 1 '12 at 23:31 Here's how to complete GEdgar's Weierstrass solution. Start with the differential equation$$(f^\prime (z))^2 = c\,f(z)^3 + f(z)^2$$and introduce a new function g(z) satisfying the relation$$f(z)=\frac4{c}g(z)-\frac1{3c}$$(This is essentially equivalent to "depressing" a cubic equation plus a rescaling.) We can then derive a differential equation for g(z) through this substitution:$$(g^\prime (z))^2=4g(z)^3-\frac1{12}g(z)+\frac1{216}$$and comparing this with the Weierstrass differential equation, we find that$$g(z)=\wp\left(z;\frac1{12},-\frac1{216}\right)$$We then compute the discriminant$$\Delta=\left(\frac1{12}\right)^3-27\left(-\frac1{216}\right)^2=0$$and find that the underlying cubic is degenerate. Abramowitz and Stegun give appropriate formulae for the case of \Delta=0, g_2 > 0, and g_3 < 0 (see formula 18.12.3); applying the formula listed there yields$$\wp\left(z;\frac1{12},-\frac1{216}\right)=\frac1{12}+\frac14\mathrm{csch}^2\left(\frac{z}{2}\right)$$and thus$$f(z)=\frac4{c}\left(\frac1{12}+\frac14\mathrm{csch}^2\left(\frac{z}{2}\right)\right)-\frac1{3c}=\frac1{c}\mathrm{csch}^2\left(\frac{z}{2}\right)$\$