Linear Second Order ODE/y'' - y' - 6 y = exp -x/Proof 1

Proof
It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients 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:
 * $y_g = C_1 e^{3 x} + C_2 e^{-2 x}$

We have that:
 * $R \left({x}\right) = e^{-x}$

and so from the Method of Undetermined Coefficients for the Exponential function:
 * $y_p = \dfrac {K e^{a x} } {a^2 + p a + q}$

where:
 * $K = 1$
 * $a = -1$
 * $p = -1$
 * $q = -6$

Hence:

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


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