Answer
$y=e^{2x}(c_{1}cos3x+c_{2}sin3x)$
Work Step by Step
$y''-4y'+13y=0$
Use auxiliary equation
$r^{2}-4r+13=0$
$r^{2}-4r+13=0$
$r=\frac{-(-4)\sqrt ((-4)^{2}-4(1)(13))}{2(1)}$
$r=\frac{4±\sqrt (16-52)}{2}$
$r=\frac{4±\sqrt (-36)}{2}$
$r=\frac{4±6i}{2}$
$r= 2±3i$
$r_{1}=2-3i$
$r_{2}=2+3i$
$α=2$, $β=3$
Formula 11
$y=e^{αx}(c_{1}cosβx+c_{2}sinβx)$
$y=e^{2x}(c_{1}cos3x+c_{2}sin3x)$