Laplace Transforms

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{;} \\ \mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{ }\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1} \\ \mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}{e}^{-t},0⩽t<3,\\ 1,3<t\end{array} \end{gathered}$$

The unit triangular pulse $\mathit{\Lambda }\mathbf{\left(}\mathit{t}\mathbf{\right)}$is defined by 

$\mathit{\Lambda }\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}0,t<0\\ 2t,0<t<1/2\\ 2-2t,1/2<t<1\\ 0,t>1\end{array}$
1.sketch the graph of  why is it named? why is its symbol appropriate. 

2. show that  $\mathit{\Lambda }\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{t}}\mathbf{2}\left\{{\Pi }_{0,1/2}\left(\tau \right)-{\Pi }_{1/2,1}\left(\tau \right)\right\}\mathbf{d}\mathit{t}$

3 Find the Laplace transform of $\mathit{\Lambda }\mathbf{\left(}\mathit{t}\mathbf{\right)}$

Determine, where the periodic function is described by its graph.

Determine , where the periodic function is described by its graph.

Determine \(L\{ f\} \), where the periodic function is described by its graph.

In Problems\(21 - 30\), determine \({\mathcal{L}^{ - 1}}\{ F\} \)

\(sF(s) + 2F(s) = \frac{{10{s^2} + 12s + 14}}{{{s^2} - 2s + 2}}\)

The mixing tank in Figure 7.18 initially holds 500 L of a brine solution with a salt concentration of 0.02 kg/L. For the first 10 min of operation, valve A is open, adding 12 L/min of brine containing a 0.04 kg/L salt concentration. After 10 min, valve B is switched in, adding a 0.06 kg/L concentration at 12 L/min. The exit valve C removes 12 L/min, thereby keeping the volume constant. Find the concentration of salt in the tank as a function of time.

Suppose in Problem 33 valve B is initially opened for10 min and then valve A is switched in for 10 min. Finally, valve B is switched back in. Find the concentration of salt in the tank as a function of time.

Suppose valve C removes only 6 L/min in Problem 33.Can Laplace transforms be used to solve the problem? Discuss.

Show that if\(L\{ g\} (s) = {\left[ {(s + \alpha )\left( {1 - {e^{ - Ts}}} \right)} \right]^{ - 1}}\), where\(T > 0\)is fixed, then

\(\begin{array}{c}g(t) = {e^{ - \alpha t}} + {e^{ - \alpha (t - T)}}u(t - T)\\ + {e^{ - \alpha (t - 2T)}}u(t - 2T)\\ + {e^{ - \alpha (t - 3T)}}u(t - 3T) + L\end{array}\)

[Hint: Use the fact that\(\left. {1 + x + {x^2} + L = 1/(1 - x).} \right]\)

In Problems 15, find a Taylor series for f(t) about t=0. Assuming the Laplace transform of f(t) can be computed term by term, find an expansion for \(\mathcal{L}\{ f\} (s)\)in powers of 1 / s. If possible, sum the series.

15. \(f(t) = {e^t}\)

In Problems 16, find a Taylor series for f(t) about t=0. Assuming the Laplace transform of f(t) can be computed term by term, find an expansion for \(\mathcal{L}\{ f\} (s)\)in powers of 1 / s. If possible, sum the series.

\({\rm{\;16}}{\rm{.\;}}f(t) = \sin t\)

In Problems 17, find a Taylor series for \(f\left( t \right)\) about t=0. Assuming the Laplace transform of \(f\left( t \right)\) can be computed term by term, find an expansion for \(\mathcal{L}\{ f\} (s)\)in powers of \(1/s\). If possible, sum the series.

\({\rm{\;17}}{\rm{.\;}}f(t) = \frac{{1 - \cos t}}{t}\)

In Problems 18, find a Taylor series for \(f\left( t \right)\) about t=0. Assuming the Laplace transform of \(f\left( t \right)\) can be computed term by term, find an expansion for \(\mathcal{L}\{ f\} (s)\)in powers of \(1/s\). If possible, sum the series.

\({\rm{\;18}}{\rm{.\;}}f(t) = {e^{ - {r^2}}}\)

Using the recursive relation (7) and the fact that \({\rm{\Gamma }}(1/2) = \sqrt \pi   \cdot \)

Determine a)\(\mathcal{L}\left\{ {{t^{ - 1/2}}} \right\}\)b)\(\mathcal{L}\left\{ {{t^{7/2}}} \right\}\)

Use the recursive relation (7) and the fact that \(\Gamma (1/2) = \sqrt \pi  \) to show that

\({L^{ - 1}}\left\{ {{s^{ - (n + 1/2)}}} \right\}(t) = \frac{{{2^n}{t^{n - 1/2}}}}{{1 \times 3 \times 5L(2n - 1)\sqrt \pi  }}\)

Where n is a positive integer. 

Use the convolution theorem to obtain a formula for the solution to the given initial value problem, where \(g(t)\)is piecewise continuous on \([0,\infty )\)and of exponential order.

\(y'' - 2y' + y = g(t);\quad y(0) =  - 1,\quad y'(0) = 1\) 

Use the convolution theorem to obtain a formula for the solution to the given initial value problem, where \(g(t)\)is piecewise continuous on \([0,\infty )\)and of exponential order.

\(y'' + 9y = g(t);\;\;\;y(0) = 1,\;\;\;y'(0) = 0\) 

Use the convolution theorem to obtain a formula for the solution to the given initial value problem, where $\mathit{g}\left(t\right)$is piecewise continuous on $\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$ and of exponential order.

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

Use the convolution theorem to obtain a formula for the solution to the given initial value problem, where g(t) is piecewise continuous on $\mathbf{1}\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$and of exponential order.

$\mathbf{[}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{ }\mathbf{+}\mathbf{ }\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathit{g}\mathbf{(}\mathit{t}\mathbf{)}\mathbf{ }\mathbf{;}\mathbf{ }\mathit{y}\mathbf{(}\mathbf{0}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{,}\mathbf{ }\mathit{y}\mathbf{\text{'}}\mathbf{(}\mathbf{0}\mathbf{)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{1}$

Use the convolution theorem to find the inverse Laplace transform of the given function.

$\frac{\mathbf{1}}{\mathbf{s}\mathbf{(}{\mathbf{s}}^{\mathbf{2}}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{1}\mathbf{)}}$

Use the convolution theorem to find the inverse Laplace transform of the given function.

$\frac{\mathbf{1}}{\mathbf{(}\mathbf{s}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{2}\mathbf{)}}$

Use the convolution theorem to find the inverse Laplace transform of the given function.

 $\frac{\mathbf{14}}{\mathbf{(}\mathbf{s}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{5}\mathbf{)}}$

In problem 15-22, solve the given integral equation or integro-differential equation for $\mathit{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$

$\mathit{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathit{L}\underset{0}{\overset{t}{\int }}(t-v)\mathit{y}\mathbf{\left(}\mathbf{v}\mathbf{\right)}\mathit{d}\mathit{v}\mathbf{=}{\mathit{t}}^{\mathbf{2}}$

Use the convolution theorem to find the inverse Laplace transform of the given function.

$\frac{\mathbf{1}}{{\left(s^2+4\right)}^{\mathbf{2}}}$

Use the convolution theorem to find the inverse Laplace transform of the given function.

$\frac{\mathbf{s}}{{\left(s^2+1\right)}^{\mathbf{2}}}$

Use the convolution theorem to find the inverse Laplace transform of the given function.

$\frac{\mathbf{1}}{{\mathbf{s}}^{\mathbf{3}}\left(s^2+1\right)}$

Use the convolution theorem to find the inverse Laplace transform of the given function.$\frac{\mathbf{s}}{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{2}\mathbf{)}}\left[Hint: \frac{{\displaystyle s}}{{\displaystyle s-1}}=1+\frac{{\displaystyle 1}}{{\displaystyle s-1}}\right]$

Find the Laplace transform of $\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\underset{0}{\overset{t}{\int }}(t-v)e^{3v}dv$

Find the Laplace transform of .$\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\underset{0}{\overset{t}{\int }}{e}^{v}sin(t-v)dv$

Solve the given integral equation or integro-differential equation for y(t) $\mathit{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{3}\underset{0}{\overset{t}{\int }}y\left(v\right)sin(t-v)dv\mathbf{=}\mathit{t}$       

Solve the given integral equation or integro-differential equation for y(t)$\mathit{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{3}\underset{0}{\overset{t}{\int }}{e}^{t-v}y\left(v\right)dv\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}$

In problem 15-22, solve the given integral equation or integro differential equation for$\mathit{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$

$\mathit{L}\underset{0}{\overset{t}{\int }}(t-v)\mathit{y}\mathbf{\left(}\mathbf{v}\mathbf{\right)}\mathit{d}\mathit{v}\mathbf{=}\mathbf{1}$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

7.

\(\begin{array}{l}(D - 4)[x] + 6y = 9{e^{ - 3t}};x(0) =  - 9\\x - (D - 1)[y] = 5{e^{ - 3t}};y(0) = 4\end{array}\)

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{l}\mathbf{D}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{7}\mathbf{/}\mathbf{4}\mathbf{,}\\ \mathbf{4}\mathbf{x}\mathbf{+}\mathbf{D}\mathbf{\left[}\mathbf{y}\mathbf{\right]}\mathbf{=}\mathbf{3}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{4}\end{array}$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{l}{\mathbf{x}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{2}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{-}\mathbf{x}\mathbf{;}\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{7}\mathbf{,}\\ \mathbf{-}\mathbf{3}{\mathbf{x}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{2}{\mathbf{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{=}\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathbf{,}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{9}\end{array}$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{lll}\mathbf{\text{x''+y=1;}}& \mathbf{\text{x(0)=1,}}& {\mathbf{\text{x}}}^{\mathbf{\text{'}}}\mathbf{\text{(0)=1}}\\ {\mathbf{\text{x+y}}}^{\mathbf{\text{''}}}\mathbf{\text{=-1;}}& \mathbf{\text{y(0)=1,}}& {\mathbf{\text{y}}}^{\mathbf{\text{'}}}\mathbf{\text{(0)=-1}}\end{array}$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{ll}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{u}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{;}& \mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\\ \mathbf{x}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}\mathbf{;}& \mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\end{array}$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{l}{\mathbf{\text{x}}}^{\mathbf{\text{'}}}\mathbf{\text{+y=x;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x(0)=0,y(0)=1}}\\ {\mathbf{\text{2x}}}^{\mathbf{\text{'}}}{\mathbf{\text{+y}}}^{\mathbf{\text{''}}}{\mathbf{\text{=u(t-3);y}}}^{\mathbf{\text{'}}}\mathbf{\text{(0)=-1}}\end{array}$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{c}{\mathbf{\text{x}}}^{\mathbf{\text{'}}}{\mathbf{\text{-y}}}^{\mathbf{\text{'}}}\mathbf{\text{=(sint)u(t-π);\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x(0)=1, }}\\ {\mathbf{\text{x+y}}}^{\mathbf{\text{'}}}\mathbf{\text{=0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y(0)=1}}\end{array}$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{ccc}{\mathbf{\text{x}}}^{\mathbf{\text{''}}}\mathbf{\text{=y+u(t-1);}}& \mathbf{\text{x(0)=1,}}& {\mathbf{\text{x}}}^{\mathbf{\text{'}}}\mathbf{\text{(0)=0}}\\ {\mathbf{\text{y}}}^{\mathbf{\text{''}}}\mathbf{\text{=x+1-u(t-1);}}& \mathbf{\text{y(0)=0,}}& {\mathbf{\text{y}}}^{\mathbf{\text{'}}}\mathbf{\text{(0)=0}}\end{array}$

use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

 ${\mathit{x}}^{\mathbf{\text{'}}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{2}$$\mathit{x}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$

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

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

x'-2x+y'= -(cost+4 sint) ;  x(π)=0

2x+y'+y=sint+3cost ;  y(π)=3

Use the method of Laplace transforms to solve

$\begin{array}{ll}{\mathbf{x}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}\mathbf{;}& \mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{3}\begin{array}{cc}\mathbf{,}& {\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\end{array}\\ \mathbf{4}\mathbf{x}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{6}\mathbf{;}& \mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathbf{.}\end{array}$

For the interconnected tanks problem of Section 5.1, page 241, suppose that the input to tank A is now controlled by a valve which for the first 5 min delivers \(6\;L/min\)of pure water, but thereafter delivers \(6\;L/min\) of brine at a concentration of \(0.02\;kg/L.\)Assuming that all other data remain the same (see Figure 5.1, page 241), determine the mass of salt in each tank for \(t > 0\)if \({x_0} = 0\)and \({y_0} = 0.04\)

Recompute the coupled mass–spring oscillator motion in Problem 1, Exercises 5.6 (page 287), using Laplace transforms.

In Problems 1 and 2, use the definition of the Laplace transform to determine $\mathit{L}\mathbf{\left\{}\mathbf{f}\mathbf{\right\}}$.

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\mathbf{3}\mathbf{,}\mathbf{0}\mathbf{\le }\mathbf{t}\mathbf{\le }\mathbf{2}\\ \mathbf{6}\mathbf{-}\mathbf{t}\mathbf{,}\mathbf{2}\mathbf{<}\mathbf{t}\end{array}$

In Problems 1 and 2, use the definition of the Laplace transform to determine $\mathit{L}\mathbf{\left\{}\mathbf{f}\mathbf{\right\}}$.

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}{\mathbf{e}}^{\mathbf{-}\mathbf{t}}\mathbf{,}\mathbf{0}\mathbf{\le }\mathbf{t}\mathbf{\le }\mathbf{5}\\ \mathbf{-}\mathbf{1}\mathbf{,}\mathbf{5}\mathbf{<}\mathbf{t}\end{array}$

In Problems 3–10, determine the Laplace transform of the given function.${\mathit{t}}^{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{9}\mathbf{t}}$

In Problems 3–10, determine the Laplace transform of the given function.${\mathit{e}}^{\mathbf{3}\mathbf{t}}\mathit{s}\mathit{i}\mathit{n}\mathbf{4}\mathit{t}$