Nonlinear System of DEs

1. The problem statement, all variables and given/known data

Solve the System of DEs:

[itex]\sqrt{1+y'^{2}+z'^{2}}-\frac{y'^{2}}{\sqrt{1+y'^{2}+z'^{2}}}=C_{1}[/itex]

[itex]\sqrt{1+y'^{2}+z'^{2}}-\frac{z'^{2}}{\sqrt{1+y'^{2}+z'^{2}}}=C_{2}[/itex]

2. Relevant equations

The two equations above are quite relevant.

3. The attempt at a solution

I attempted basic substitution to do is this:
Multiply through by the radical
Cancel terms
Solve for y' and z' in terms of each other
Plug them into each other and then attempt to solve

I ended up trying to solve for y' first. What I got is a solvable polynomial in terms of y'; a quite tedious looking polynomial at that. I stopped here and began erasing. Maybe I was doing it right, but I don't even want to see what happens when I plug it into the quadratic equation and then attempt to substitute it back into the z' equation.. It sounds like WAYYY to long of a process considering this is physics homework, not math homework.

Can anybody give me some advice? Any good way to approach problems like these? I'm starting to encounter them a lot and it's always the part of the problem that I spend hours looking at :\

 

Ok thanks. That was a pretty straightforward step.

This is what I did:
Plugged Q into the original system.
Solved the system for y'^2 and z'^2
Plugged my values for y'^2 and z'^2 into the expression for Q
Solved for Q

Q again come in the form of a quadratic equation with two solutions. The solutions were both in terms of C1 and C2. At this point I used to following logic to prove that y and z are both linear.

Q is a constant, thus, y' and z' squared are constants, thus, y' and z' are constants, thus, y and z are linear.

Considering that there were two values for Q though, is this the correct way of thinking about it? Do the two Q values basically say that there are two solutions of y and z, both solutions being linear? As long as I can prove the solutions are linear then I've successfully answered the question.

 

Thanks, I appreciate the help :)

I'm looking forward to experimenting with these nonlinear systems. It's a shame we never covered them in my ODE class.