How to solve this nonlinear wave-like equation?

Question

The equation has a form as: $$ \frac{d^2 q}{dt^2} + f(t) q = g(t) $$

The concrete equation is : $$ 1.61504 \frac{d^2 q}{dt^2} + (13.6833 + 0.312247 \dot{\theta}^2)q= 7.34943 \ddot{\theta} $$ $$ \dot{\theta}= \frac{4}{15}t - \frac{4}{2\pi}\sin(\frac{2\pi}{15}t) $$ I have seen some papers about solution to nonlinear wave-like eqaution. But I don't get the solution.

If you can solve it or have read papers or websites about the solution including numerical solution, please help me.

Answer 1

I have tried numerical solution by using ode45 functionin MATLAB. Aha. Maybe it dosen't have analytical solution. Awww！

Here is the MATALB code:

tspan = [0,15];
y0 = [0,0.01];
[t,y] = ode45(@(t,y) odefun(t,y),tspan,y0);
plot(t,y(:,1),'-o',t,y(:,2),'-.')
plot(t,y(:,1))

function dydt = odefun(t,y)
dydt = zeros(2,1);
dydt(1) = y(2);
dydt(2) = 1.21349 - 1.21349 *cos(2*pi*t/15)-( 8.5116 + 0.0137484*t^2 - 0.0391781*cos((4 *pi* t)/15) - 0.0656436*t*sin(2 *pi* t)/15)*y(1);
end

The solution of q (Picture)