Chapter 7: The Laplace Transform

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

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathit{t}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{1}$

$\mathit{L}\left\{\underset{0}{\overset{t}{\int }}{e}^{aT}f\left(T\right)dT\right\}\mathbf{=}\underset{\mathbf{¯}}{}$ 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 , whose graph is given in figure 7.R.1

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

26.$f\left(t\right)={\left(2t-1\right)}^3$

Find theLaplace transform $\mathrm{f}\times \mathrm{g}$of the given functions. Do not evaluate convolutionintegral before transforming where $\mathbf{L}\left\{e^{2t}*\mathrm{sint}\right\}$

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

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}{\mathit{t}}^{\mathbf{3}}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}$

Question:

$\mathit{L}\mathbf{\left\{}\underset{\mathbf{0}}{\overset{\mathbf{t}}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{a}\mathbf{T}}\mathbf{f}\left(T\right)\mathbf{d}\mathbf{T}\mathbf{\right\}}\mathbf{=}\underset{\mathbf{¯}}{}$ 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 19–36 use theorem 7.1.1 to find$L\left\{f\left(t\right)\right\}.$

27.$f\left(t\right)=1+e^{4t}$

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

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

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

27.$\mathit{y}\mathbf{-}\mathbf{6}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{13}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{3}$

Question:

$\mathit{L}\left\{\underset{0}{\overset{t}{\int }}{e}^{aT}f\left(T\right)dT\right\}\mathbf{=}\underset{\mathbf{¯}}{}$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$y=f\left(t\right)$ , whose graph is given in figure 7.R.1