Answer to Question #233578 in Differential Equations for sam

Question

Find the general solution of the equation y^''− 4y^'+ 13y = 0.

Answer 1

Let us find the general solution of the differential equation $y''− 4y'+ 13y = 0$. The characteristic equation $k^2-4k+13=0$ is equivalent to $(k-2)^2=-9,$ and hence has the roots $k_1=2+3i$ and $k_2=2-3i.$ Therefore, the general solutions is of the form:

$y=e^{2x}(C_1\cos (3x)+C_2\sin(3x)).$