You need to use matrices to solve this system of differential equations.
First you need to divide both by "m" so you get x'' and y'' on the left.
Then you express these together as a matrix equation:
[x'' y'']^T=[A][x' y']
The matrix A should be formed so that when this is expanded you get the same result as your initial equations.
You then need to find the eigenvalues and eigenvectors of the matrix A. This will then lead you to your general solution which will include a lot of exponents! Then you can substitute in your initial conditions.
This should hopefully push you in the right direction!