Chapter 7: The Laplace Transform

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

38.$f\left(t\right)={\cos }^{2}t$

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

$\mathit{L}\left\{e^{2-t}\mathcal{U}(t-2)\right\}$

Consider the differential equation ${\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{x}\mathbf{)}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{(}\mathit{x}\mathbf{+}\mathbf{2}\mathbf{)}\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{.}$

Verify that${\mathit{y}}_{\mathbf{1}}\mathbf{=}\mathit{x}$is one solution of the associated homogeneous equation. Then show that the method of reduction of order discussed in Section 4.2 leads to a second solution${\mathit{y}}_{\mathbf{2}}$ of the homogeneous equation as well as a particular solution ${\mathit{y}}_{\mathbf{p}}$of the nonhomogeneous equation. Form the general solution of the DE on the interval$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$

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

Equation.

38. $\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{W}\mathbf{h}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}{t}^2,0⩽t<1\\ 0,t⩾1,\end{array}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$

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

39. $f\left(t\right)=\sin \left(4t+5\right)$

$L\left\{tU\right(t-2\left)\right\}$

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

equation.

39.$\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}$$\mathbf{w}\mathbf{h}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{}\mathbf{i}\mathbf{s}\mathbf{g}\mathbf{i}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{i}\mathbf{n}\mathbf{F}\mathbf{i}\mathbf{g}\mathbf{u}\mathbf{r}\mathbf{e}\mathbf{7}\mathbf{.}\mathbf{R}\mathbf{.}\mathbf{10}\mathbf{.}\mathbf{}$

In Problems 39–44 solve the given differential equation subject to the indicated conditions.

$\mathit{y}\mathbf{"}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{(}\mathbf{\pi }\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\left(}\mathbf{\pi }\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}$

In Problems 1–18 use Definition 7.1.1 to find $L\left\{f\left(t\right)\right\}.$

3. $f\left(t\right)={\{}_{1,t⩾1}^{t,0⩽t<1}$

Use the Laplace transform to solve the given initial- value problem.

$\mathbf{}\mathbf{3}\mathbf{.}\mathbf{}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{\delta }\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{\pi }\mathbf{)}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$