ajedrezlaproa6j

2021-12-30

Solve the given initial-value problem.

Timothy Wolff

Find the homogeneous solution of the given IVP as follows.
The given IVP is as follows,
and $\omega \ne \gamma$
and $\omega \ne \gamma$
Consider the homogeneous part $x{}^{″}+{\omega }^{2}x=0$
The characterice equation is ${m}^{2}+{\omega }^{2}=0$
$⇒{m}^{2}+{\omega }^{2}=0$
$⇒=-\omega$
$⇒=-\omega i,\omega i$
The homogeneous solution is ${x}_{h}={C}_{1}\mathrm{cos}\left(\omega t\right)+{C}_{2}\mathrm{sin}\left(\omega t\right)$
Find the particular solution of the given IVP as follows.
The IVP is and $\omega \ne \gamma$
Let the particular solution be of the form, ${x}_{p}=A\mathrm{cos}\left(\gamma t\right)$
${x}^{\prime }=-\gamma A\mathrm{sin}\left(\gamma t\right)$
$x{}^{″}=-{\gamma }^{2}\mathrm{cos}\left(\gamma t\right)$
$⇒-{\gamma }^{2}\mathrm{cos}\left(\gamma t\right)+{\omega }^{2}A\mathrm{cos}\left(\gamma t\right)={F}_{0}\mathrm{cos}\left(\gamma t\right)$
$⇒A{\omega }^{2}-{\gamma }^{2}={F}_{0}$
$⇒A=\frac{{F}_{0}+{\gamma }^{2}}{{\omega }^{2}},\omega \ne 0$
The particular solution is ${x}_{p}=\frac{{F}_{0}+{\gamma }^{2}}{{\omega }^{2}}\mathrm{cos}\left(\gamma t\right)$
The general solution is of the given IVP is as follows.
The general solution is
$x\left(t\right)={x}_{h}+{x}_{p}$
$={C}_{1}\mathrm{cos}\left(\omega t\right)+{C}_{2}\mathrm{sin}\left(\omega t\right)+\frac{{F}_{0}+{\gamma }^{2}}{{\omega }^{2}}\mathrm{cos}\left(\gamma t\right)$

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