Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x
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Theorem
The second order ODE:
- $(1): \quad y - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$
has the general solution:
- $y = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x} + \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$
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:
- $p = -2$
- $q = -5$
- $\map R x = 2 \cos 3 x - \sin 3 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
- $y - 2 y' - 5 y = 0$
From Linear Second Order ODE: $y - 2 y' - 5 y = 0$, this has the general solution:
- $y_g = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x}$
We have that:
- $\map R x = 2 \cos 3 x - \sin 3 x$
and it is noted that $2 \cos 3 x - \sin 3 x$ is not itself a particular solution of $(2)$.
We then determine the particular solution:
Particular Solution
The second order ODE:
- $(1): \quad y - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$
has a particular solution:
- $y_p = \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$
So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:
- $y = y_g + y_p = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x} + \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$
$\blacksquare$