Linear Second Order ODE/y'' - y' - 6 y = 0

Theorem

The second order ODE:

$(1): \quad y - y' - 6 y = 0$

has the general solution:

$y = C_1 e^{3 x} + C_2 e^{-2 x}$

Proof

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

Its auxiliary equation is:

$(2): \quad: m^2 - m - 6 = 0$

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

$m_1 = 3$

$m_2 = -2$

These are real and unequal.

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

$y = C_1 e^{3 x} + C_2 e^{-2 x}$

$\blacksquare$