Chapter 7: The Laplace Transform

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

2.$\mathit{f}\left(\mathit{t}\right)=\left\{\begin{array}{c}4,0\le t<2\\ 0,t\ge 2\end{array}\right.$

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

$\mathbf{}\mathbf{2}\mathbf{.}\mathbf{}{\mathbf{y}}^{\mathbf{r}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{\delta }\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}$

in problems 1-20 find either f(S)or f(t),as indicated $\mathit{L}\left\{t{e}^{-6t}\right\}$

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

30.$f\left(t\right)={\left(e^t-e^{-t}\right)}^2$

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

$\mathbf{L}\left\{{\int }_0^t\mathrm{TsinTdT}\right\}$

In problem 21-30use the Laplace transform to solve the givenInitial-value Problem.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{1}\mathbf{+}\mathit{t}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{4}$

In Problems 29–32 express f in terms of unit step functions. Find$\mathit{L}\left\{f\left(t\right)\right\}$

and$\mathit{L}\left\{e^{t}f\left(t\right)\right\}$.

30.

Use the procedures developed in this chapter to find the general solution of each differential equation.

$\mathbf{2}{\mathit{x}}^{\mathbf{3}}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{19}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{39}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathbf{0}$

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

31.$f\left(t\right)=4t^2-5\sin 3t$

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

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