# Solution of non-linear second order differential equation

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?

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

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, $$\int \frac{dy}{\sqrt{y_0^2-\frac45(1+y_0)^{5/2}+\frac45(1+y)^{5/2}}}=t+C$$
• 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 Nov 23, 2017 at 12:33