Chapter 7: The Laplace Transform

Consider a battery of constant voltage ${\mathit{E}}_{\mathbf{0}}$ that charges the capacitor shown in Figure 7.3.10. Divide equation (20) by L and define $\mathbf{2}\mathit{\lambda }\mathbf{=}\mathit{R}\mathbf{/}\mathit{L}$ and ${\mathit{\omega }}^{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{/}\mathit{L}\mathit{C}$.Use the Laplace transform to show that the solution $\mathit{q}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{\lambda }\mathit{q}\mathbf{\text{'}}\mathbf{+}{\mathit{\omega }}^{\mathbf{2}}\mathit{q}\mathbf{=}{\mathit{E}}_{\mathbf{0}}\mathbf{/}\mathit{L}$q(t) of subject to$\mathit{q}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{i}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$ is

$q\left(t\right)=\left\{\begin{array}{l}{E}_{0}C\left[1-e^{-\lambda t}\left(\cosh \sqrt{{\lambda }^2-{\omega }^{2}t}\right.\right.\left.\left.+\frac{\lambda }{\sqrt{{\lambda }^2-{\omega }^2}}\sinh \sqrt{{\lambda }^2-{\omega }^2}t\right)\right]\lambda >\omega \\ E_{0}C\left[1-e^{-\lambda t}\left(\cos \sqrt{{\omega }^2-{\lambda }^2}t\right.\right.\left.\left.+\frac{\lambda }{\sqrt{{\omega }^2-{\lambda }^2}}\sin \sqrt{{\omega }^2-{\lambda }^2}t\right)\right]\lambda <\omega \end{array}\right.$

Given that $\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$ is a solution of ${\mathit{y}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{11}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{\text{,}}$

find the general solution of the DE without the aid of a calculator or a computer.

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

36.$f\left(t\right)=e^{-t}\cosh t$

Use the Laplace Transform to find the charge q(t) in an RC series circuit when$\mathit{q}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$ and$\mathit{E}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{E}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{kt}}\mathbf{,}\mathit{k}\mathbf{>}\mathbf{0}$

Consider two cases:$\mathit{k}\mathbf{\ne }\mathbf{1}\mathbf{/}\mathit{R}\mathit{C}$ and$\mathit{k}\mathbf{\ne }\mathbf{1}\mathbf{/}\mathit{R}\mathit{C}$.

Find a linear second-order differential equation with constant coefficients for which ${\mathit{y}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$ and ${\mathit{y}}_{\mathbf{2}}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{x}}$are solutions of the associated homogeneous equation and ${\mathit{y}}_{\mathbf{p}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathit{x}$is a particular solution of the non homogeneous equation.

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

Equation.

36.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{8}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{20}\mathbf{y}\mathbf{=}\mathbf{t}{\mathbf{e}}^{\mathbf{t}}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 37–40 find $L\left\{f\left(t\right)\right\}$by first using a trigonometric identity.

37.$f\left(t\right)=\sin 2t\cos 2t$

In problems 37-48 find either F(s) or f(t), as indicated.

$\mathit{L}\mathbf{\left\{}\mathbf{\right(}\mathit{t}\mathbf{-}\mathbf{1}\mathbf{\left)}\mathit{U}\mathbf{\right(}\mathit{t}\mathbf{-}\mathbf{1}\mathbf{\left)}\mathbf{\right\}}$

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

Equation.

37.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{-}\mathbf{t}\mathbf{u}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

(a) Write the general solution of the fourth-order DE ${\mathit{y}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathbf{-}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{0}$entirely in terms of hyperbolic functions.

(b) Write down the form of a particular solution of${\mathit{y}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathbf{-}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathit{x}$