Solving Nonlinear ode in mathematica

Question

I want to solve n ODE in Mathematica, but there is an error I cannot understand: "The function x appears with no arguments."

This is the code I used:

eqn = x''[y] + a x'[y] Sin[y - y0] + a x[y] (Cos[y - y0] - s) == 0; 

sol = DSolve[eqn, x, y]

Could anyone help me? Is there another function that solve the problem?

Answer 1

DSolve is going to have issues with your ode, I doubt that there is a closed form answer. You might look instead at using NDSolve and settle for a numerical approximation. Notice that this will force you to pick concrete values for $y_0$ in order to run numerical solvers.

Alternatively, you could linearize the equation and use DSolve to capture leading order behavior (just like we approximate a pendulum by $y'' + y = 0$ instead of $y'' + \sin y = 0$). Techniques exist to extend this method to capture higher order terms instead of just linearizing.

Answer 2

Continuation of the answer of @erfink: Use NDSolve and see what influence parameters have.

 eqn[y0_, xs_, x0_, s_, a_] = 
       x''[y] + a x'[y] Sin[y - y0] + a x[y] (Cos[y - y0] - s) == 0 && 
        x'[0] == xs && x[0] == x0;

xsol[z_, y0_, xs_, x0_, s_, a_, b_, 
c_] := (x[z] /. First@NDSolve[eqn[y0, xs, x0, s, a], x, {y, b, c}])

Manipulate[
Plot[xsol[z, y0, xs, x0, s, a, -3, 3], {z, -3, 3}], {{y0, 0}, -3, 
3}, {{xs, 0}, -3, 3}, {{x0, 1}, -3, 3}, {{s, 0}, -3, 
3}, {{a, 1}, -3, 3}]

Attention: Use z instead of y. Otherwise you get problems with the y in NDSolve.