Linear Second Order ODE/y'' + 4 y' + 5 y = 0

Theorem

The second order ODE:

$(1): \quad y + 4 y' + 5 y = 0$

has the general solution:

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

Proof

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

Its auxiliary equation is:

$(2): \quad m^2 + 4 m + 5 = 0$

From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:

$m_1 = -2 + i$

$m_2 = -2 - i$

So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:

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

$\blacksquare$