Solution of Constant Coefficient Homogeneous LSOODE

Theorem
Let:
 * $(1): \quad y'' + p y' + q y = 0$

be a constant coefficient homogeneous linear second order ODE.

Let $m_1$ and $m_2$ be the roots of the auxiliary equation $m^2 + p m + q = 0$.

Then $(1)$ has the general solution:


 * $y = \begin{cases}

C_1 e^{m_1 x} + C_2 e^{m_2 x} & : p^2 > 4 q \\ & \\ C_1 e^{m_1 x} + C_2 x e^{m_2 x} & : p^2 = 4 q \\ & \\ e^{a x} \paren {C_1 \sin b x + C_2 \cos b x} & : p^2 < 4 q \end{cases}$ where:
 * $a + i b = m_1$
 * $a - i b = m_2$