Linear Second Order ODE/2 y'' + 2 y' + 3 y = 0

Theorem
The second order ODE:
 * $(1): \quad 2 y'' + 2 y' + 3 y = 0$

has the general solution:
 * $y = e^{-x/2} \left({C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}\right)$

Proof
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.

Its auxiliary equation is:
 * $(2): \quad: 2 m^2 + 2 m + 3 = 0$

From Solution to Quadratic Equation: Real Coefficients, the roots of $(2)$ are:
 * $m_1 = - \dfrac 1 2 + \dfrac {\sqrt 5} 2 i$
 * $m_2 = - \dfrac 1 2 - \dfrac {\sqrt 5} 2 i$

These are complex and unequal.

So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
 * $y = e^{-x/2} \left({C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}\right)$