Linear Second Order ODE/y'' - 2 y' - 5 y = 0

Theorem
The second order ODE:
 * $(1): \quad y'' - 2 y' - 5 y = 0$

has the general solution:
 * $y = C_1 \, \map \exp {\paren {1 + \sqrt 6} x} + C_2 \, \map \exp {\paren {1 - \sqrt 6} x}$

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

Its auxiliary equation is:
 * $(2): \quad: m^2 - 2 m - 5 = 0$

From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
 * $m_1 = 1 + \sqrt 6$
 * $m_2 = 1 - \sqrt 6$

So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
 * $y = C_1 \, \map \exp {\paren {1 + \sqrt 6} x} + C_2 \, \map \exp {\paren {1 - \sqrt 6} x}$