Why DSolve[] does not work?

Question

I try to find the analytical solution to the ordinary differential equations as fellow:

$$\frac{d^2 \theta}{d t^2} = \dfrac{-M\cos\theta\sin\theta(\frac{d\theta}{dt})^2 - \frac{g}{l}\sin\theta}{1-M\cos^2\theta}$$ $$\frac{d^2 x}{d t^2} = \dfrac{Mg\cos\theta\sin\theta - Ml\sin\theta(\frac{d\theta}{dt})^2}{1-M\cos^2\theta}$$

And my code is

eqn1 = D[y[t], {t,2}] - (-M Cos[y[t]] Sin[y[t]] (y'[t])^2 - g/l Sin[y[t]])/(1 -M (Cos[y[t]])^2);
eqn2 = D[x[t], {t, 2}] - (M g Sin[y[t]] Cos[y[t]] + M l Sin[y[t]] (y'[t])^2) /(1 - M (Cos[y[t]])^2);
eqnSet = {eqn1 == 0, eqn2 == 0};
DSolve[eqnSet, {y[t], x[t]}, t]

DSolve does not give a solution, instead it writes:

DSolve[{-((-((g Sin[y[t]])/l) - 
  M Cos[y[t]] Sin[y[t]] Derivative[1][y][t]^2)/(
 1 - M Cos[y[t]]^2)) + (y^\[Prime]\[Prime])[t] == 0, -((g M Cos[y[t]] Sin[y[t]] + 
  l M Sin[y[t]] Derivative[1][y][t]^2)/(1 - M Cos[y[t]]^2)) + (
 x^\[Prime]\[Prime])[t] == 0}, {y[t], x[t]}, t]

could anyone point out my problem?

Answer 1

This is a non linear equation which does not seem to have a closed form solution.

Try instead something like this:

eqn1 = D[y[t], {t, 
     2}] - (-M Cos[y[t]] Sin[y[t]] (y'[t])^2 - g/l Sin[y[t]])/(1 -   M (Cos[y[t]])^2);
eqn2 = D[x[t], {t, 
     2}] - (M g Sin[y[t]] Cos[y[t]] + M l Sin[y[t]] (y'[t])^2)/(1 -  M (Cos[y[t]])^2);
eqnSet = {eqn1 == 0, eqn2 == 0, y[0] == 1, x[0] == 2, y'[0] == 1, 
   x'[0] == 1};
sol=NDSolve[eqnSet /. {M -> 1 , g -> 1, l -> 1}, {y[t], x[t]}, {t, 0, 1}]

Then for instance

ParametricPlot[{x[t], y[t]} /. sol[[1]], {t, 0, 1},AspectRatio -> 1]