Definition:Steady-State/Second Order ODE

Definition

Consider the second order ODE:

$\dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$

where:

$K \in \R_{>0}$

$a, b \in \R_{>0}: b < a$

which has the general solution:

$(1): \quad y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$

where:

$\alpha = \sqrt {a^2 - b^2}$

$\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$

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

Also see

• Definition:Transient

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems