Definition:Steady-State/Second Order ODE

Definition
Consider the second order ODE:
 * $\dfrac {\mathrm d^2 y} {\mathrm d x^2} + 2 b \dfrac {\mathrm d y} {\mathrm d x} + a^2 x = K \cos \omega x$

where:
 * $K \in \R: k > 0$
 * $a, b \in \R_{>0}: b < a$

which has the general solution:
 * $(1): \quad y = e^{-b x} \left({C_1 \cos \alpha x + C_2 \sin \alpha x}\right) + \dfrac K {\sqrt {4 b^2 \omega^2 + \left({a^2 - \omega^2}\right)^2} } \cos \left({\omega x - \phi}\right)$

where:
 * $\alpha = \sqrt {a^2 - b^2}$
 * $\phi = \arctan \left({\dfrac {2 b \omega} {a^2 - \omega^2} }\right)$

The term $\dfrac K {\sqrt {4 b^2 \omega^2 + \left({a^2 - \omega^2}\right)^2} } \cos \left({\omega x - \phi}\right)$ is known as the steady-state component of $(1)$.

Also see

 * Definition:Transient