Here's an outline of the solution. Write the equations as

or in vector form .

Now diagonalise that matrix . You should find that the eigenvalues are –1 and –6, with corresponding eigenvectors and . Let P be the matrix having those eigenvectors as columns, and let D be the diagonal matrix with the eigenvalues –1 and –6 as its diagonal elements. Then and so .

Let . Then and . Those are simple harmonic motion equations, with solutions , .

Finally, , giving the solution