Answer
y = $C_1cos(\sqrt 2x)$ + $C_2sin(\sqrt 2x)$
Work Step by Step
Question: y"+2y=0
We know:
y = $e^{rx}$
y' = $re^{rx}$
y" = $r^{2}e^{rx}$
This results in $(r^{2}+2)e^{rx}$=0
The corresponding characteristic equation is:
$r^{2}+2=0$
Solving the equation with complex numbers gives:
$r^{2}=-2$
$r=\frac{+}{-}\sqrt 2 i$
Adding the constants gives the following general solution:
y = $C_1cos(\sqrt 2x)$ + $C_2sin(\sqrt 2x)$
