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:

$y = e^{-x/2} \paren {C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}$

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 with 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:

$y = e^{-x/2} \paren {C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}$

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.17$: Problem $1 \ \text{(g)}$