How to solve this couple of second ODEs

Question

I try to solve this couple of second order differential equations to get an analytic solution for x(t) and y(t)

(x'[t]/x[t]) (y'[t]/3*y[t]) + y''[t]/3*y[t] + 2 x''[t]/x[t] =0

2*(x'[t]/x[t]) y'[t] + y''[t] = 0  

I'm a beginner in MA. However I think DSolve can not solve these coupled ODE in one step, here is my trial :

eq1[t_] := x'[t]/x[t] * y'[t]/(3 y[t]) + y''[t]/(3 y[t]) +  2 *x''[t]/x[t]
eq2[t_] := 2 x'[t] /x[t] y'[t] + y''[t]
DSolve[{eq1[t] == eq2[t] == 0, x[0] == 0, y[0] == 0}, {x, y}, t]

Which gives no solution. Noting that the IC are arbitrary. So any help how to go around that, I mean can the function x given in terms of y from eq1, then substitute in eq2 which will be in this case a function of y only.

Answer 1

Your initial conditions: x[0]==0, y[0]==0 are not compatible with your equations that contain terms like: 1/x[t]and 1/y[t]. Further a second order DE needs 2 initial conditions for a specific solution, you need to add the initial values of the derivative.

You may try to solve these equations numerically by choosing suitable initial values. E.g.:

eq1[t_] := 
 x'[t]/x[t]*y'[t]/(3 y[t]) + y''[t]/(3 y[t]) + 2*x''[t]/x[t]
eq2[t_] := 2 x'[t]/x[t] y'[t] + y''[t]
sol[t_] = {x[t], y[t]} /. 
   NDSolve[{eq1[t] == 0, eq2[t] == 0, x[0] == 1, y[0] == 2, 
      x'[0] == 1, y'[0] == -1}, {x, y}, {t, 0, 5}][[1]];
Plot[sol[t], {t, 0, 10}, PlotRange -> {0, 5}]