Solving ODE involving square of first derivative

[tsw99]

Homework Statement

this is not from a math course, but from Gregory's classical mechanics book prob 2.10
it's easy to obtain the desired ODE
$$\dot{r}^{2}=\frac{u^{2}}{a^{2}}(\frac{U^{2}a^{2}}{a^{2}}-r^{2})$$
since it's non-linear, i have a difficult time to solve for r(t)
u, U and a are some constants with unit speed, speed and length resp.

Homework Equations

The Attempt at a Solution

all methods i know fail, I believe there is some trick that I am not aware of. great appreciate for any help:(

 

[hunt_mat]

The equation you have can be written as:
$$\dot{r}=\pm\frac{u}{a}\sqrt{U^{2}-r^{2}}$$
Dividing and integrating shows that:
$$\int\frac{dr}{\sqrt{U^{2}-r^{2}}}=\pm\frac{u}{a}\int dt$$
The integral can be solved by the substitution:
$$r=U\sin\alpha$$
I will leave you to slog through the algerbra.

 

[tsw99]

The equation you have can be written as:
$$\dot{r}=\pm\frac{u}{a}\sqrt{U^{2}-r^{2}}$$
Dividing and integrating shows that:
$$\int\frac{dr}{\sqrt{U^{2}-r^{2}}}=\pm\frac{u}{a}\int dt$$
The integral can be solved by the substitution:
$$r=U\sin\alpha$$
I will leave you to slog through the algerbra.

oh...thank you very much!
I think i need to brush up my math skills...edit: in fact i type the ODE wrongly, but the method should be similar