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