How to solve differential equation of 3 order analytically

I would like to solve a system of differential equaions

x''[t] == -a0*(a1 - b*z'[t])*cos[w*t], x[to] == 0, x'[to] == 0
z''[t] == -a0*b*x'[t]*cos[w*t], z[to] == 0, z'[to] == 0

It reduces to a third order equation

z'''[t] == a*(1-c*z'[t])*cos^2[w*t]-tan(w*t)*w*z''[t], z[to] == 0, z'[to] == 0

syms u(x) a c  w d
Du = diff(u, x);
D2u = diff(u, x, 2); 
u(x) = dsolve(diff(u, x, 3) == a*(1-c*Du)*(cos(x))^2-tan(x)*d*D2u, u(to) == 0, Du(to) == 0)
Does not give solution. How to solve it in steps, maybe first for z'?

Answer 1

[Star Strider]

I would keep it as the original system (and change ‘to’ to 0):

syms x(t) z(t) a0 a1 b w 
Dz = diff(z);
D2z = diff(z,2);
Dx = diff(x);
D2x = diff(x,2);
Eq1 = D2x == -a0 * (a1 - b*Dz) * cos(w*t);
Eq2 = D2z == -a0 * b * Dx * cos(w*t);
Soln = dsolve(Eq1, Eq2, x(0) == 0, Dx(0) == 0, z(0) == 0, Dz(0) == 0);
X = Soln.x;
Z = Soln.z;
X = simplify(X, 'steps', 20)
Z = simplify(Z, 'steps', 20)

This gives you two ‘solutions’ involving integrals, that it transformed with dummy variable ‘y’:

X =
-(a1*int(sin((a0*b*sin(w*y))/w), y, 0, t))/b

Z =
-(a1*int(exp(-(a0*b*sin(w*y)*1i)/w)*(exp((a0*b*sin(w*y)*1i)/w) - 1)^2, y, 0, t))/(2*b)

This is likely as close as you can get to an analytic solution.