I am working with a system, which is described by the following differential equation

$y'' = (1+y)^{3/2}$

Does this differential equation have a closed-form solution? And more generally: Is there a place, which contains a complete list of second-order differential equations, which do have analytical solutions?

  • $\begingroup$ Check the solution on Wolfram Alpha - it has an extremely messy solution that uses the hypergeometric function. Have you made a typo somewhere? $\endgroup$
    – Toby Mak
    Nov 23, 2017 at 10:15
  • $\begingroup$ Anyway do you have the initial conditions or any other restriction on your system? This could make the integration a lot easier. $\endgroup$ Nov 23, 2017 at 10:23
  • $\begingroup$ Boundary conditions are y(0)=0, y(L)=a. $\endgroup$
    – user13514
    Nov 23, 2017 at 12:26

1 Answer 1


Multiply with $2y'$ and integrate once to get $$ y'^2-y_0'^2 = \frac45(1+y)^{5/2}-\frac45(1+y_0)^{5/2} $$ This gives you the curves in the phase plane that the solutions follow, any further integration leads to moderately nasty integrals, $$ \int \frac{dy}{\sqrt{y_0^2-\frac45(1+y_0)^{5/2}+\frac45(1+y)^{5/2}}}=t+C $$

  • $\begingroup$ The boundary conditions are y(0)=0 and y(L)=a, where a and L are arbitrary constants. $\endgroup$
    – user13514
    Nov 23, 2017 at 12:25
  • $\begingroup$ Then you have to solve $$\int_0^a\frac{dy}{\sqrt{2E+\frac45(1+y)^{5/2}}}=L$$ for $E$, where $2E=y'^2-\frac45(1+y)^{5/2}$ is a constant of motion, you can call $E$ the energy. Which leads you into the realm of special functions $\endgroup$ Nov 23, 2017 at 12:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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