Answer
$y=5cos2x+3sin2x$
Work Step by Step
$y''+4y=0$
Use auxiliary equation
$r^{2}+4=0$
$r^{2}=-4$
$r_{1}=2i$
$r_{2}=-2i$
$r_{1}=α+βi$
$r_{2}=α-βi$
$α=0$
$β=2$
$y=e^{αx}(c_{1}cosβx+c_{2}sinβx)$
$y=c_{1}cos2x+c_{2}sin2x)$
$y(0)=5$
$5=c_{1}cos2(0)+c_{2}sin2(0))$
$5=c_{1}(1)+c_{2}(0)$
$c_{1}=5$
$y(\frac{\pi}{4})=3$
$3=c_{1}cos2(\frac{\pi}{4})+c_{2}sin2(\frac{\pi}{4})$
$3=c_{1}(0)+c_{2}(1)$
$c_{2}=3$
$y=5cos2x+3sin2x$
