Solving the differential equation $y'' + 2y' + 2y = 0$ given constraints How can I solve this initial value problem?
$$ y'' + 2y' + 2y = 0,$$ given $y\,(\pi/4)=2$ and $y'(\pi/4)=0$.
I've found $y(t)=e^{-t} \left(C_1\cos t + C_2\sin t \right)$ but I wasn't able to find $C_1$ and $C_2$. How can I find them?

Using the $y(t)$ you found (which is correct), substitute $\dfrac{\pi}{4}$ into $y(t)$, yielding:
$$\dfrac{1}{\sqrt{2}}e^{-\pi/4}(c_1 + c_2) = 2$$
Substitute $\dfrac{\pi}{4}$ into $y'(t)$, yielding:
$$-\sqrt{2}e^{-\pi/4} c_1 = 0$$
This gives us $c_1 = 0$.
Can you now find $c_2$?
