Chapter 7: The Laplace Transform

A$\mathbf{4}$-Pound weight stretches a spring$\mathbf{2}$feet. the weight is released from rest$\mathbf{18}$inches above the equilibrium position, and the resulting motion takes place in a medium offering a damping force numerically equal to$\frac{\mathbf{7}}{\mathbf{8}}$times the instantaneous velocity. use the Laplace transform to find the equation of motion x(t).

In Problems 33 and 34 sketch the graph of the given function. Find $\mathit{L}\left\{f\left(t\right)\right\}$

.$\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{+}\mathbf{2}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{k}\mathbf{+}\mathbf{1}}\mathbf{u}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{k}\mathbf{)}$

Write down the form of the general solution $\mathit{y}\mathbf{=}{\mathit{y}}_{\mathbf{c}}\mathbf{+}{\mathit{y}}_{\mathbf{p}}$v of the given differential equation in the two cases $\mathit{\omega }\mathbf{\ne }\mathit{\alpha }$and $\mathit{\omega }\mathbf{=}\mathit{\alpha }$. Do not determine the coefficients in
${\mathbf{y}}_{\mathbf{p}}$.

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}{\mathit{\omega }}^{\mathbf{2}}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha }\mathit{x}$

In Problems 19–36 use theorem 7.1.1 to find$L\left\{f\left(t\right)\right\}.$

34.$f\left(t\right)=\cosh kt$

Find the Laplace transform of f * g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming.

$\mathbf{L}\left\{t{\int }_0^t{\mathrm{Te}}^{-T}\mathrm{dT}\right\}$

Recall that the differential equation for the instantaneous charge q(t) on the capacitor in an LRC-series circuit is given by$\mathit{L}\frac{{\mathbf{d}}^{\mathbf{2}}}{\mathbf{d}{\mathbf{t}}^{\mathbf{2}}}\mathbf{+}\mathit{R}\frac{\mathbf{d}\mathbf{q}}{\mathbf{d}\mathbf{t}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{C}}\mathit{q}\mathbf{=}\mathit{E}\mathbf{\left(}\mathit{t}\mathbf{\right)}$,

See section 5.1. Use the Laplace transform to find q(t) when$\mathit{L}\mathbf{=}\mathbf{1}\mathit{h}\mathbf{,}\mathit{R}\mathbf{=}\mathbf{20}\mathit{\Omega }\mathbf{,}\mathit{C}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{005}\mathit{f}\mathbf{,}\mathit{E}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{150}\mathit{V}\mathbf{,}\mathit{t}\mathbf{>}\mathbf{0}\mathbf{,}\mathit{q}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{a}\mathit{n}\mathit{d}\mathit{i}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$. What is the current i(t)?

In Problems 35–42 use the Laplace transform to solve the given

equation $\mathbf{f}\left(t\right)\underset{\mathbf{k}\mathbf{=}\mathbf{0}}{\overset{\mathbf{=}}{\mathbf{=}\mathbf{\sum }\left(2k+1-t\right)}}\mathbf{\left[}\mathbf{9}\mathbf{\right(}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{k}\mathbf{)}\mathbf{-}\mathbf{4}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{k}\mathbf{-}\mathbf{1}\mathbf{\left)}\mathbf{\right]}$

write down the form of the general solution $\mathit{y}\mathbf{=}{\mathit{y}}_{\mathbf{c}}\mathbf{+}{\mathit{y}}_{\mathbf{p}}$v of the given differential equation in the two cases $\mathit{\omega }\mathbf{\ne }\mathit{\alpha }$and $\mathit{\omega }\mathbf{=}\mathit{\alpha }$. Do not determine the coefficients in ${\mathit{y}}_{\mathbf{p}}$.

role="math" localid="1667912855985" ${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}{\mathit{\omega }}^{\mathbf{2}}\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{\alpha }\mathbf{x}}$

In Problems 19–36 use theorem 7.1.1 to find $L\left\{f\left(t\right)\right\}.$

35.$f\left(t\right)=e^t\sinh t$

Use (8) to evaluate inverse Laplace transform.

${\mathrm{L}}^{-1}\left\{\frac{1}{\mathrm{s}\left(\mathrm{s}-1\right)}\right\}$