Nonlinear System of DEs

[Hertz]

Homework Statement

Solve the System of DEs:

$\sqrt{1+y'^{2}+z'^{2}}-\frac{y'^{2}}{\sqrt{1+y'^{2}+z'^{2}}}=C_{1}$

$\sqrt{1+y'^{2}+z'^{2}}-\frac{z'^{2}}{\sqrt{1+y'^{2}+z'^{2}}}=C_{2}$

Homework Equations

The two equations above are quite relevant.

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 :\

 

[UltrafastPED]

Create a new variable: $Q^{2}={1+y'^{2}+z'^{2}}$ and carry out the replacement.

This gives you a new system which looks easier - solve for Q and z; then apply the auxiliary equation above so that you can go back to y and z.

 

[Hertz]

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.

 

[UltrafastPED]

The substitution generated an additional set of values because Q could be + or -; since your original problem used square root, which implies the + branch, you could just rule out the - by that reason.

Think of the extra solutions as a bonus - you learned something, but you only have to turn in the + half!

BTW this is "multiplication" of solutions is common with the method of substitution - though the method really cuts down the work!

 

[Hertz]

The substitution generated an additional set of values because Q could be + or -; since your original problem used square root, which implies the + branch, you could just rule out the - by that reason.

Think of the extra solutions as a bonus - you learned something, but you only have to turn in the + half!

BTW this is "multiplication" of solutions is common with the method of substitution - though the method really cuts down the work!

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.

 

[UltrafastPED]

Nonlinear is an advanced topic; systematic techniques for finding solutions are
Lacking IIRC.