Here is the equation I typed
DSolve[{((x''[t])/(2*x[t]))+((y[t])/(4*(x[t])^3))+(((x'[t])^2)/(4*(x[t])^2))-
((y''[t])/(2*y[t]))-(((y'[t])^2)/(4*(y[t]^2)))==0,
((3*x''[t])/(2*x[t]))-(((x'[t])^2)/(4*(x[t])^2))-((y[t])/(2*(x[t])^3))+((y''[t])/(2*y[t]))-
((3*((y'[t])^2))/(4*(y[t]^2)))+((2)/(x[t]*y[t]))-((x'[t]*y'[t])/(x[t]*y[t]))==0},{x,y},t]