Answer
$y(x)=e^{3x}+e^{-x}$
Work Step by Step
You substitute the $y'', y'$ and $y$ with $y''=r^{2}e^{rx}, y'=re^{rx}$ and $y=e^{rx}$. And by filling out the equation you can find the final equation.
After this you use the given $y(0)=a$ and $y'(0)=b$ in order to find the two constants.
