Linear Second Order ODE/y'' + 3 y' - 10 y = 6 exp 4 x

Theorem
The second order ODE:
 * $(1): \quad y'' + 3 y' - 10 y = 6 e^{4 x}$

has the general solution:
 * $y = C_1 e^{2 x} + C_2 e^{-5 x} + \dfrac {e^{4 x} } 3$

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:
 * $p = 3$
 * $q = -10$
 * $R \left({x}\right) = 6 e^{4 x}$

First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
 * $y'' + 3 y' - 10 y = 0$

From Second Order ODE: $y'' + 3 y' - 10 y = 0$, this has the general solution:
 * $y_g = C_1 e^{2 x} + C_2 e^{-5 x}$

We have that:
 * $R \left({x}\right) = 6 e^{4 x}$

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:
 * $K = 6$
 * $a = 4$
 * $p = 3$
 * $q = -10$

Hence:

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


 * $y = y_g + y_p = C_1 e^{2 x} + C_2 e^{-5 x} + \dfrac {e^{4 x} } 3$