Nonlinear Differential Equation

**Assassin0071** · October 13th 2012, 04:32 PM

The Question is:

Solve the following equation. The constant $\mu$ is non negative and $h$ is greater than zero.

${ ( \frac{dr}{dt} ) }^2 = \frac{2\mu}{r} + 2h$

Any help would be apperciated as I'm not entirely sure where to start with this problem.

**MaxJasper** · October 13th 2012, 06:54 PM

Most likely, solution looks like:

$\frac{\mu \text{Log}\left[h \sqrt{r}+\sqrt{h} \sqrt{\mu +h r}\right]}{h^{3/2}}-\frac{\sqrt{r} \sqrt{\mu +h r}}{h} = \pm \sqrt{2} t+\text{C1}$

**JJacquelin** · October 14th 2012, 07:43 AM

Hi Assassin0071 !
In fact, it is an ODE with separable variables :
dr/dt = sqrt((2m/r)+2h) = sqrt((2m+2hr)/r)
dt = dr/sqrt(r/(2m+2hr))
Then, integrate it.

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