# General solution for system of equations with complex eigenvalues

How do I find the general solution for the system of equations:

$\dot{x}=-y$

$\dot{y}=x$

with $x(0)=1$ and $y(0)=1$. I've found the characteristic equation to be $\lambda^2=-1\implies \lambda_{1,2}=+-i$. I know the general solution of a system of equations takes the form $c_1\vec{x}_1+c_2\vec{x}_2$, but I'm not sure exactly where to go. Thanks for any help

• You may like to obtain a second order ODE in y and then solve. – Nitin Uniyal Nov 29 '15 at 4:41
• The general solution to $\dot{\mathbf v}=A\mathbf v$ is $\exp(tA)$. Do you know how to find the exponential of a matrix with complex eigenvalues? – amd Nov 29 '15 at 6:04