# System of Implicit Non-Linear First Order ODEs

1. Oct 7, 2015

### CSteiner

I have an extremely messy system of differential equations. Can anyone offer any ideas for a general solution?

p(t) is a function of t, and A is a constant.

2. Oct 7, 2015

### SlowThinker

An idea would be to rewrite $\cos \arctan \frac{\dot{y}}{\dot{x}}$ as $\frac{\dot x}{\sqrt{\dot x^2+\dot y^2}}$, and similar for the other equation. Then try to simplify it a bit.

3. Oct 7, 2015

### CSteiner

Didn't think of that, thanks! I'll keep working on it.

4. Oct 8, 2015

### pasmith

You may be best served by switching to intrinsic coordinates $(s, \psi)$ where $$\dot x = \dot s \cos\psi, \\ \dot y = \dot s \sin \psi.$$ Your system is then $$(\dot s - p) \cos \psi = 0 \\ (\dot s - p) \sin \psi = -At.$$ Now either $\dot s = p$ or $\cos \psi = 0$. The first is impossible unless $A = 0$. The second requires that $\dot x = 0$ and $\dot y = \pm p - At$.

If $A = 0$ then $\dot s = p$ and $\psi$ can be an arbitrary function of $t$.

5. Oct 8, 2015

### CSteiner

Sorry, but it looks like something went wacky with your latex code. Could you retype it? It's hard to understand what your saying.

6. Oct 8, 2015

### SlowThinker

It takes time before equations render correctly but eventually they do. What are you seeing? Try a different browser or device.
Basically he says to substitute $\dot x = \dot s \cos\psi$, $\dot y = \dot s \sin \psi$. Then it's very simple to arrive at a solution.

BTW, perhaps the first solution should have been $\dot x=0$, $\dot y=p\pm At$.