ODE2.2

$2y''+y'+4y=0\,$

Set up and solve the characteristic equation.

$2m^2+m+4=0\,$

$m = \frac{-1 \pm \sqrt{1-4\cdot8}}{4} =\frac{-1}{4} \pm \frac{i\sqrt{31}}{4}\,$

So the general solution is

$y(x) = c_1e^{-1/4x}\cos\left(\frac{\sqrt{31}}{4}x\right) + c_2e^{-1/4x}\sin\left(\frac{\sqrt{31}}{4}x\right)\,$