Chapter 7: The Laplace Transform

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

28.$f\left(t\right)=t^2-e^{-9t}+5$

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

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

In problems 21-30 use the Laplace transform to solve the given initial-value problem.

28. $\mathbf{2}\mathit{y}\mathbf{-}\mathbf{20}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{51}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

$\mathit{L}\left\{\underset{0}{\overset{t}{\int }}{e}^{aT}f\left(T\right)dT\right\}\mathbf{=}$ whereas$\mathit{L}\left\{e^{at}\underset{0}{\overset{t}{\int }}f\left(T\right)dT\right\}\mathbf{=}\underset{\mathbf{¯}}{}$

In Problems 25–28 use the unit step function to and an equation for each graph in terms of the function $\mathit{y}\mathbf{=}\mathit{f}\left(t\right)$, whose graph is given in figure 7.R.1

In Problems 17-32 use the procedures developed in this chapter to find the general solution of each differential equation.

$\mathit{y}\mathbf{"}\mathbf{-}\mathit{y}\mathbf{=}\frac{\mathbf{2}{\mathbf{e}}^{\mathbf{x}}}{{\mathbf{e}}^{\mathbf{x}}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}}$

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

22.$f\left(t\right)={\left(1+e^{2t}\right)}^2$

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

$\mathbf{L}\left\{{\int }_0^t{e}^{-T}\mathrm{cosTdT}\right\}$

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

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{t}}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}$

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

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

29.

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

$\mathbf{6}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{5}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{-}\mathit{y}\mathbf{=}\mathbf{0}$