Solve ODEs w/ Mathematica & Compute Derivative of Niles

[Niles]

Hi

I have a system of ODEs of the form

dx/dt = v
dv/dt = a = C*f(x),

where C denotes a constant and f(x) is some function of x. This system is easy to solve using (e.g.)

NDSolve[x''[t] == C*f(x), x[0] == 0, x'[0] == 0}, x, {t, 0, tMax}];

I need to use the derivative of the solution x[t], x'[t], in the following expression: B(x) = A + v(x), where A denotes a constant. But please note that the derivative is needed as a function of x, not t. I've been trying to figure out a smart way to do this, but I can't wrap my head around this. What should I do to achieve this?Niles.

 

[phyzguy]

Since you have x[t], you need to invert this to find t[x], then, since you know v[t], your v[x] is given by v[t[x]]. If you have an analytic solution, you can do this analytically, but if you have a numerical solution, the easiest way to invert it is probably with FindRoot. See the attached notebook.

 

[Niles]

Thanks, that is very kind of you.Niles.