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:

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

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:

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

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.17$: Problem $1 \ \text{(a)}$