Solve the system of differential equations: $$ \frac{\mathrm{d}}{\mathrm{d}t}x=2y, \quad \frac{\mathrm{d}}{\mathrm{d}t}y=2x, \quad \frac{\mathrm{d}}{\mathrm{d}t}z=2x $$
I tried solving the first two equations by eliminating $y$ and solving for $x$, which gives $$x(t)=c_1e^{2t}+c_2e^{-2t},$$ but the answer is somewhat different. Please help me.