How to solve a coupled differential equations I tried different ways to solve this differential equation but I did not succeed. These is the first couple ODEs I try to solve. I hope somebody can give me a hint.
\begin{eqnarray}
\ddot{x} + ax - b\dot{y} = c
\end{eqnarray}
\begin{eqnarray}
\ddot{y} + ay + b\dot{x} = 0
\end{eqnarray}

Go to the complex plane. Do "first" + i"second" equation and you'll get something elegant. Sum of equations with second multiplied by the imaginary unit:
$$(\ddot{x}+i\ddot{y})+a(x+iy)+b(-\dot{y}+i\dot{x})=c+0i$$
Use new complex variable $z=x+iy$:
$$\ddot{z}+a z +ib\dot{z}=c$$
This is a simple second order DE with known solutions. Even better, the particular solution is just $z=c/a$ and the homogenous part works out very elegantly with $z=Ce^{i\lambda t}$, which gets you to a completely real characteristic polynomials ($i$ multiplies out):
$$-\lambda^2-\lambda b+a=0$$
Of course, $\lambda$-s may still be a complex conjugate pair, but the fact that the polynomial is real simplifies everything.
