Nonlinear DE depending on two variables

[KingBongo]

I have a single nonlinear differential equation like

F(x2,dx1/dt,dx2/dt,d2x1/dt2,d2x2/dt2)=0

where x1=x1(t), x2=x2(t)

i.e. a second order non-linear DE with no implicit dependence on x1(t). I suppose solving it results (in general) that one of the variables would be dependent on the other.

How can I proceed with this?

 

[Office_Shredder]

Do you have an actual equation to solve? If so, posting it would probably help

 

[KingBongo]

Ok. I will use dJxi for dJxi/dtJ, i \in {1,2} and so on. Little bit ugly but anyway, :)

(d1x1)^3 +(d1x2)^2*d1x1 -x2*d1x1*d2x2 +x2*d2x1*d1x2 +L*x2*((d1x1)^2+(d1x2)^2)^(3/2) =0

This comes from a calculus of variations problem, L is a Lagrange coefficient, and the DE describes what the extremals should look like.

I have some boundary constraints, x1(t0)=0, x2(t0)=R, x2(t1)=r, R≥0, r≥0 and an integral constraint that has to hold, that's why I use a Lagrange coefficient. I started a thread regarding this in some other section as well.

This problem was meant to be an exercise before the real problem I am working on, :) I thought it would be easy to solve but no.

 

[Office_Shredder]

lol... that's a bit out of my league.

But I can do the latex!

$$(\frac{dx_1}{dt})^3 + (\frac{dx_2}{dt})^2 \frac{dx_1}{dt} - x_2\frac{dx_1}{dt}\frac{d^2x_2}{dt^2} + x_2\frac{d^2x_1}{dt^2}\frac{dx_2}{dt} + Lx_2( (\frac{dx_1}{dt})^2 + (\frac{dx_2}{dt})^2 )^{3/2} = 0$$

 

[KingBongo]

NO! It is simple, solve it! :) Thanks.