Linear Second Order ODE/y'' - y = 3 exp -x

Theorem
The second order ODE:
 * $(1): \quad y'' - y = 3 e^{-x}$

has the general solution:
 * $y = C_1 e^x + C_2 e^{-x} - \dfrac {3 x e^{-x} } 2$

Proof
It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
 * $y'' + p y' + q y = \map R x$

where:
 * $p = 0$
 * $q = -1$
 * $\map R x = 3 e^{-x}$

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

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

We have that:
 * $\map R x = 3 e^{-x}$

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

where:
 * $K = 3$
 * $a = -1$
 * $p = 0$

Hence:

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


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