Linear Second Order ODE/y'' - y' - 6 y = exp -x/Proof 2

Proof
It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
 * $y'' + p y' + q y = R \left({x}\right)$

where:
 * $p = -1$
 * $q = -6$
 * $R \left({x}\right) = e^{-x}$

First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
 * $y'' - y' - 6 y = 0$

From Second Order ODE: $y'' - y' - 6 y = 0$, this has the general solution:
 * $y_g = C_1 e^{3 x} + C_2 e^{-2 x}$

It remains to find a particular solution $y_p$ to $(1)$.

Expressing $y_g$ in the form:
 * $y_g = C_1 y_1 \left({x}\right) + C_2 y_2 \left({x}\right)$

we have:

By the Method of Variation of Parameters, we have that:


 * $y_p = v_1 y_1 + v_2 y_2$

where:

where $W \left({y_1, y_2}\right)$ is the Wronskian of $y_1$ and $y_2$.

We have that:

Hence:

It follows that:

So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:


 * $y = y_g + y_p = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$

is the general solution to $(1)$.