Find the solution of the given initial value problem.
$2y''+y'-4y=0,$
$y(0)=0$
$y'(0)=1$
$2r^2+r-4$
$r=\frac{-1\pm\sqrt{1-4(2)(-4)}}{4}$
$r_{1}=\frac{-1+\sqrt{33}}{4}$
$r_{2}=\frac{-1-\sqrt{33}}{4}$
$y=c_{1}e^{-\frac{-1+\sqrt{33}}{4}}+c_{2}e^{\frac{-1-\sqrt{33}}{4}}$
Sub $y(0)=0$ and we get:
$c_1+c_2=0$
Take derivative:
$y'=-\frac{-1-\sqrt{33}}{4}c_{1}e^{-(\frac{1-\sqrt{33}}{4})t}-\frac{1+\sqrt{33}}{4}c_{2}^{-(\frac{1+\sqrt{33}}{4})t}$
Sub $y'(0)=1$ into $y'$:
$1=-\frac{1-\sqrt{33}}{4}c_{1}-\frac{1+\sqrt{33}}{4}c_{2}$
Sub $c_{1}=-c_{2}$ into the above equation:
$-\frac{1-\sqrt{33}}{4}(-c_{2})-\frac{1+\sqrt{33}}{4}c_{2}=1$
$c_{2}(\frac{1-\sqrt{33}}{4}-\frac{1+\sqrt{33}}{4})=1$
$c_{2}(\frac{-2\sqrt{33}}{4})=1$
$c_{2}=-\frac{4}{2\sqrt{33}}$
Since $c_{1}=-c_{2}$
$c_{1}=\frac{4}{2\sqrt{33}}$
$\therefore y=\frac{4}{2\sqrt{33}}e^{\frac{-1+\sqrt{33}}{4}t}-\frac{4}{2\sqrt{33}}e^{\frac{-1-\sqrt{33}}{4}t}$
