Answer
$y=\frac{e^{4-2x}-e^{2x}}{e^{4}-1}$
Work Step by Step
$y''=4y$
Use auxiliary equation
$r^{2}-4=0$
$r^{2}=4$
$r=±2$
$r_{1}=-2$
$r_{2}=2$
Formula 8
$y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}$
$y=c_{1}e^{-2x}+c_{2}e^{2x}$
$y(0)=1$
$1=c_{1}e^{-2(0)}+c_{2}e^{2(0)}$
$1=c_{1}+c_{2}$
$y(1)=0$
$0=c_{1}e^{-2(1)}+c_{2}e^{2(1)}$
$0=c_{1}+c_{2}e^{4}$
$1=c_{1}+c_{2}$
$0=c_{1}+c_{2}e^{4}$
Solve system of equations
$1=c_{2}(1-e^{4})$
$c_{2}=\frac{1}{1-e^{4}}$
$c_{1}=1-\frac{1}{1-e^{4}}=\frac{-e^{4}}{1-e^{4}}=\frac{e^{4}}{e^{4}-1}$
$y=\frac{e^{4} \times e^{-2x}-e^{2x}}{e^{4}-1}$
$y=\frac{e^{4-2x}-e^{2x}}{e^{4}-1}$
