Answer
$y=c_{1}cos(\frac{3x}{5})+c_{2}sin(\frac{3x}{5})$
Work Step by Step
$25y''+9y=0$
Use auxiliary equation
$25r^{2}+9=0$
$25r^{2}+9=0$
$25r^{2}=-9$
$r^{2}=-\frac{9}{25}$
$r=±\sqrt (-\frac{9}{25})$
$r=±\frac{3}{5} \times \sqrt (-1)$
$r= ±\frac{3}{5}i$
$r_{1}=0+\frac{3}{5}i$
$r_{2}=0-\frac{3}{5}i$
$α=0$
$β=\frac{3}{5}$
By Formula 11,
$y=e^{αx}(c_{1}cosβx+c_{2}sinβx)$
$y=c_{1}cos(\frac{3x}{5})+c_{2}sin(\frac{3x}{5})$
