Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have to solve the following differential equation: $$\frac{d^2\phi(x)}{dx^2}=\phi(x)^2$$ The solution is given by:

$$\phi(x)=6\wp(x+c_1,0,c_2)$$ where $c_1,c_2$ are constants. I have the following boundary conditions: $$\phi(-L)=\phi(L)=\phi_0$$ How can I solve the previous ODE taking into account the conditions at the boundaries? Thank you in advance.

share|cite|improve this question
up vote 1 down vote accepted

$$\phi''= \phi^2 $$ $$\phi'\phi''= \phi'\phi^2 $$

$$\int \phi'\phi'' dx=\int \phi^2 \phi'dx$$

$$\frac{\phi'^{2}}{2} = \frac{\phi^3}{3} +k $$

$$ \phi'^{2} = \frac{2 \phi^3}{3} +2k = \frac{2 \phi^3}{3} +c_1 $$

$$\phi' = \sqrt{\frac{2 \phi^3}{3} +c_1} $$

$$\frac{\phi'}{\sqrt{\frac{2 \phi^3}{3} +c_1}} =1 $$ $$\int \frac{\phi'}{\sqrt{\frac{2 \phi^3}{3} +c_1}} dx=\int dx$$

$$\int \frac{1}{\sqrt{\frac{2 \phi^3}{3} +c_1}} d\phi=x+c_2$$

I asked the integral to the wolfram and it includes elliptic integral of the first kind . The solution is here from wolfram integrator. Sorry it seems that there is no easy expression of the solution for that ODE.

You will need to take the solution from wolfram and to put the boundary conditions into that complex equation that wolfram offered and then you need to find $c_1,c_2$. If there is a simplier expression of the solution I would like to learn it.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.