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2021-12-30

Solve the given initial-value problem.

$\frac{{d}^{2}x}{{dt}^{2}}={\omega}^{2}x={F}_{0}\mathrm{cos}\left(\gamma t\right),\text{}x\left(0\right)=0,\text{}{x}^{\prime}\left(0\right)=0$

Answer & Explanation

Timothy Wolff

Expert

2021-12-31Added 26 answers

Find the homogeneous solution of the given IVP as follows.

The given IVP is as follows,

$\frac{{d}^{2}x}{{dt}^{2}}={\omega}^{2}x={F}_{0}\mathrm{cos}\left(\gamma t\right),\text{}x\left(0\right)=0,\text{}{x}^{\prime}\left(0\right)=0$ and $\omega \ne \gamma$

$\Rightarrow x{}^{\u2033}+{\omega}^{2}x={F}_{0}\mathrm{cos}\left(\gamma t\right),\text{}x\left(0\right)={x}^{\prime}\left(0\right)=0$ and $\omega \ne \gamma$

Consider the homogeneous part$x{}^{\u2033}+{\omega}^{2}x=0$

The characterice equation is${m}^{2}+{\omega}^{2}=0$

$\Rightarrow {m}^{2}+{\omega}^{2}=0$

$\Rightarrow =-\omega$

$\Rightarrow =-\omega i,\omega i$

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$\Rightarrow x{}^{\u2033}+{\omega}^{2}x={F}_{0}\mathrm{cos}\left(\gamma t\right),\text{}x\left(0\right)={x}^{\prime}\left(0\right)=0$ and $\omega \ne \gamma$

Let the particular solution be of the form,${x}_{p}=A\mathrm{cos}\left(\gamma t\right)$

${x}^{\prime}=-\gamma A\mathrm{sin}\left(\gamma t\right)$

$x{}^{\u2033}=-{\gamma}^{2}\mathrm{cos}\left(\gamma t\right)$

$\Rightarrow -{\gamma}^{2}\mathrm{cos}\left(\gamma t\right)+{\omega}^{2}A\mathrm{cos}\left(\gamma t\right)={F}_{0}\mathrm{cos}\left(\gamma t\right)$

$\Rightarrow A{\omega}^{2}-{\gamma}^{2}={F}_{0}$

$\Rightarrow A=\frac{{F}_{0}+{\gamma}^{2}}{{\omega}^{2}},\omega \ne 0$

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

$x\left(t\right)={x}_{h}+{x}_{p}$

$={C}_{1}\mathrm{cos}\left(\omega t\right)+{C}_{2}\mathrm{sin}\left(\omega t\right)+\frac{{F}_{0}+{\gamma}^{2}}{{\omega}^{2}}\mathrm{cos}\left(\gamma t\right)$

The given IVP is as follows,

Consider the homogeneous part

The characterice equation is

The homogeneous solution is

Find the particular solution of the given IVP as follows.

The IVP is

Let the particular solution be of the form,

The particular solution is

The general solution is of the given IVP is as follows.

The general solution is

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