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

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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:

$y = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x}$

$\blacksquare$