# A not so simple Second order differential equation

I need to solve the following differential equation

eqn = y''[t] - a *Cos[x[t]]*x''[t] - a *Sin[x[t]]*(x'[t])^2 == 0;


with initial conditions y[0]=0 and y'[0]=0. And y[t]=a Sin[x[t]] is the connection between the two variables x and y.

• What issues with DSolve you have? It's pretty simple. May 22, 2017 at 17:45
• You have two dependent variables, x and y. You need two equations. May 22, 2017 at 17:45
• @Bob Hanlon. You are right. I have just added that equation to my question. Could you kindly suggest how to solve this system?
– Jee
May 22, 2017 at 18:00
• Oh yes. Sorry for the poor notation. x is the function of time. Thanks @David
– Jee
May 22, 2017 at 18:16
• So you eqn is eqn = y''[t] - a*Cos[x[t]]*x''[t] - a*Sin[x[t]]*(x'[t])^2 == 0 /. y -> Function[t, a Sin[x[t]]] // FullSimplify ? May 22, 2017 at 18:19

Your equation only has a constant solution. You can see this by replacing for y:
eqn /. y -> ( a Sin[x[#]] &)

So your differential equation actually is $\sin(x) \left(\frac{\partial^2x}{\partial t^2}\right)^2=0$. The only way to satisfy this equation is $x=const$.