given \(|\ddot {x}\relax (t) \rangle +M|x\relax (t) \rangle =0\), find the eigenvectors and eigenvalues of \(M\). Then \(\Phi =\left [ V_{2},V_{2}\right ] \) is \(2\times 2\) matrix, transformation matrix. where each column is the eigenvector of \(M\). Then \(|X\relax (t) \rangle =\Phi ^{T}|x\relax (t) \rangle \) and \(|x\relax (t) \rangle =\Phi \) \(|X\relax (t) \rangle \). The new system becomes \(|\ddot {X}\left ( t\right ) \rangle +\Omega |X\relax (t) \rangle =0\) where \(\Omega \) is now diagonal matrix with eigenvalues of \(M\) on the diagonal. Solve using this. First transform initial conditions to \(X\relax (t) \). Then trandform solution back to \(|x\relax (t) \rangle \) using \(|x\relax (t) \rangle =\Phi \) \(|X\relax (t) \rangle \).