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: :$p = 2 b$ :$q = a^2$ :$\map R x = K \cos \omega x$ First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE: :$y'' + 2 b y' + a^2 y = 0$ From Linear Second Order ODE: $y'' + 2 b y' + a^2 y = 0: b < a$, this has the general solution: :$y_g = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$ Substituting $\alpha = \sqrt {a^2 - b^2}$, this can be written more compactly as: :$y_g = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x}$ We have that: :$\map R x = K \cos \omega x$ It is noted that $K \cos \omega x$ is not itself a particular solution of $(2)$. So from the Method of Undetermined Coefficients for Sine and Cosine: :$y_p = A \sin \omega x + B \cos \omega x$ where $A$ and $B$ are to be determined. Hence: {{begin-eqn}} {{eqn | l = y_p       | r = A \sin \omega x + B \cos \omega x       | c =  }} {{eqn | ll= \leadsto       | l = {y_p}'       | r = \omega A \cos \omega x - \omega B \sin \omega x       | c = Derivative of Sine Function, Derivative of Cosine Function }} {{eqn | ll= \leadsto       | l = {y_p}''       | r = -\omega^2 A \sin \omega x - \omega^2 B \cos \omega x       | c = Power Rule for Derivatives }} {{end-eqn}} Substituting into $(1)$: {{begin-eqn}} {{eqn | l = -\omega^2 \paren {A \sin \omega x + B \cos \omega x} + 2 b \omega \paren {A \cos \omega x - B \sin \omega x} + a^2 \paren {A \sin \omega x + B \cos \omega x}       | r = K \cos \omega x       | c =  }} {{eqn | ll= \leadsto       | l = -\omega^2 A \sin \omega x - 2 b \omega B \sin \omega x + a^2 A \sin \omega x       | r = 0       | c = equating coefficients }} {{eqn | l = -\omega^2 B \cos \omega x + 2 b \omega A \cos \omega x + a^2 B \cos \omega x       | r = K \cos \omega x       | c =  }} {{eqn | ll= \leadsto       | l = \paren {a^2 - \omega^2} A - 2 b \omega B       | r = 0       | c =  }} {{eqn | l = \paren {a^2 - \omega^2} B + 2 b \omega A       | r = K       | c =  }} {{eqn | ll= \leadsto       | l = \frac {2 b \omega B} {a^2 - \omega^2}       | r = A       | c =  }} {{eqn | ll= \leadsto       | l = \paren {a^2 - \omega^2} B + 2 b \omega \frac {2 b \omega B} {a^2 - \omega^2}       | r = K       | c =  }} {{eqn | ll= \leadsto       | l = B \paren {\frac {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2} {a^2 - \omega^2} }       | r = K       | c =  }} {{eqn | ll= \leadsto       | l = K \frac {a^2 - \omega^2} {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2}       | r = B       | c =  }} {{eqn | ll= \leadsto       | l = \frac {2 b \omega} {a^2 - \omega^2} K \frac {a^2 - \omega^2} {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2}       | r = A       | c =  }} {{eqn | ll= \leadsto       | l = K \frac {2 b \omega} {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2}       | r = A       | c =  }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = y_p       | r = K \frac {2 b \omega} {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2} \sin \omega x + K \frac {a^2 - \omega^2} {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2} \cos \omega x       | c =  }} {{eqn | r = \frac K {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2} \paren {2 b \omega \sin \omega x + \paren {a^2 - \omega^2} \cos \omega x}       | c =  }} {{end-eqn}} From Multiple of Sine plus Multiple of Cosine: Cosine Form: {{begin-eqn}} {{eqn | o =        | r = 2 b \omega \sin \omega x + \paren {a^2 - \omega^2} \cos \omega x       | c =  }} {{eqn | r = \sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} \map \cos {\omega x + \map \arctan {\dfrac {-2 b \omega} {a^2 - \omega^2} } }       | c =  }} {{eqn | r = \sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} \map \cos {\omega x - \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} } }       | c = Tangent Function is Odd }} {{end-eqn}} and so: {{begin-eqn}} {{eqn | l = y_p       | r = \frac {K \sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } {\paren {a^2 - \omega^2}^2 + 4 b^2 \omega^2} \map \cos {\omega x - \phi}       | c = where $\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$ }} {{eqn | r = \frac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}       | c =  }} {{end-eqn}} So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution: :$y = y_g + y_p = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$ where: :$\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$ {{qed}} Category:Examples of Constant Coefficient LSOODEs Category:Examples of Method of Undetermined Coefficients 8784espr5qejyaogjyeirfzv6k804c4