Chapter 16

(a) Define the Laplace transform \(L\\{\mathrm{f}(\mathrm{t})\\}\) for a real valued function \(\mathrm{f}(\mathrm{t})\) (b) State conditions under which \(L\\{f(t)\\}\) exists. (c) Show that \(\mathrm{L}\left\\{\mathrm{f}_{1}(\mathrm{t})\right\\}\) exists where $$ \mathrm{f}_{1}(\mathrm{t})=2 \mathrm{te}^{(\mathrm{t}) 2} \cos \mathrm{e}^{(\mathrm{t}) 2} $$ but that \(\mathrm{f}_{1}\) does not satisfy all conditions in (b).

(a) Prove that \(\mathrm{f}(\mathrm{t})=\mathrm{t}^{\mathrm{n}}, \mathrm{n}>0\), is of exponential order \(\alpha\) on \([0, \infty]\) for all \(\alpha>0\). (b) Prove that \(\mathrm{f}(\mathrm{t})=\sin \mathrm{kt}\) is of exponential order \(\alpha\) on \([0, \infty]\) for all \(\alpha>0\)

(a) Prove that \(\mathrm{f}(\mathrm{t})=\mathrm{e}^{\mathrm{at}} \sin \mathrm{kt}\) is of exponential order a on \([0, \infty] .\) (b) Prove that \(\mathrm{e}^{(\mathrm{t}) 2}\) is not of any exponential order on the interval \([c, \infty]\) for an arbitrary \(c\).

Determine those values of \(s\) for which the improper integral $$ \mathrm{G}(\mathrm{s})={ }^{\infty} \int_{0} \mathrm{e}^{-\mathrm{st}} \mathrm{dt} $$ converges, and find the Laplace transform of \(\mathrm{f}(\mathrm{t})=1\).

Find the Laplace transform of $$ f(t)=t^{n} $$ where \(\mathrm{n}\) is a positive integer.

Find the Laplace transforms of (a) \(\mathrm{f}(\mathrm{t})=\mathrm{e}^{\mathrm{kt}}\), where \(\mathrm{k}\) is a complex constant of the form \(\mathrm{k}=\operatorname{Re}\\{\mathrm{k}\\}+\mathrm{i} \operatorname{Im}\\{\mathrm{k}\\}\) with \(\operatorname{Re}\\{k\\}\) the real part of \(k, \operatorname{Im}\\{k\\}\) the imaginary part of \(k\), and \(\mathrm{i} \equiv \sqrt{(-1)}\) Use this Laplace transform to find the Laplace transforms of \(f(t)=e^{-k t} \quad\) and \(\quad f(t)=1\) (b) \(\mathrm{f}(\mathrm{t})=\sin \mathrm{kt}\) where \(\mathrm{k}\) is a real constant.

Prove the following properties of the Laplace transform denoted by \(L\\{f(t)\\}\) (a) \(L\left\\{c_{1} f_{1}(t)+c_{2} f_{2}(t)+\ldots+c_{n} f_{n}(t)\right\\}\) \(=c_{1} L\left\\{f_{1}(t)\right\\}+c_{2} L\left\\{f_{2}(t)\right\\}+\ldots+c_{n} L\left\\{f_{n}(t)\right\\}\) where all \(c_{j}\) are constants. (b) \(L\left\\{f^{(n)}(t)\right\\}=s^{n} L\\{f(t)\\}-{ }^{n} \sum_{k}=1 s^{k-1} f^{(n-k)}(C\) if \(\mathrm{f}^{(\mathrm{k})}(\mathrm{t})\) are of some finite exponential orders for \(\mathrm{k}=1,2, \ldots, \mathrm{n}-1\) and if \(L\left\\{f^{(n)}(t)\right\\}\) exists. (c) \(L\left\\{e^{-a t} f(t)\right\\}=G(s+a)\) where \(\mathrm{G}(\mathrm{s})=\mathrm{L}\\{\mathrm{f}(\mathrm{t})\\}\) and a is a real constant. (d) \(L\left\\{t^{n} f(t)\right\\}=(-1)^{n}\left[\left(d^{n} F\right) / d s^{n}\right]\) where \(\mathrm{F}(\mathrm{s})=\mathrm{L}\\{\mathrm{f}(\mathrm{t})\\}\) (e) \(L\\{(1 / t) f(t)\\}={ }^{\infty} \int_{S} F(\sigma) d \sigma\) where $$ \mathrm{F}(\mathrm{s})=\mathrm{L}\\{\mathrm{f}(\mathrm{t})\\} $$

Use the Laplace transform of $$ f(t)=e^{k t} $$ where \(\mathrm{k}\) is a complex constant of the form \(\mathrm{k}=\operatorname{Re}\\{\mathrm{k}\\}+\mathrm{i} \operatorname{lm}\\{\mathrm{k}\\}\) with \(\operatorname{Re}\\{k\\}\) the real part of \(k, \operatorname{lm}\\{k\\}\) the imaginary part of \(k\), and $$ \mathrm{i} \equiv \sqrt{(-1)} $$ to find the Laplace transforms of $$ f(t)=\cosh k t, \sinh k t, \cos \mathrm{kt}, \text { and } \sin k t $$

Find the Laplace transform, \(L\\{f(t)\\}=F(s)\), of (a) \(f(t)=2 \sin t+3 \cos 2 t\) (b) \(g(t)=\left[\left(1-e^{-t}\right) / t\right]\).

Find the Laplace transforms of (a) \(g(t)=e^{-2 t} \sin 5 t\) (b) \(\mathrm{h}(\mathrm{t})=\mathrm{e}^{-\mathrm{t}} \mathrm{t} \cos 2 \mathrm{t}\)

Find the Laplace transform of (a) \(g(t)=t e^{4 t}\) (b) \(\mathrm{f}(\mathrm{t})=\mathrm{t}^{7 / 2}\)

Find the Laplace transform $$ \mathrm{L}\\{(\sin 3 \mathrm{t}) / \mathrm{t}\\} $$

Prove that $$ L\\{f(t-a) \alpha(t-a)\\}=e^{-a s} L\\{f(t)\\} $$ for any function \(\mathrm{f}\) which has a Laplace transform, and where $$ \begin{array}{lr} a(t)=0, & t<0 \\ \text { and }=1, & t \geq 0 \end{array} $$ is the unit step function, and \(\mathrm{a}>0\).

Find the Laplace transform \(L\\{g(t)\\}\), where \(g(t)=0\) \(t<4\) and \(=(\mathrm{t}-4)^{2}, \mathrm{t} \geq 4\)

Find the Laplace transform of the function \(\mathrm{f}(\mathrm{t})\) shown in the accompanying figure and defined by \(f(t)=t, \quad 04\)

Find the Laplace transform \(L\\{f(t)\\}\) of the function shown in Figure 1 and defined by $$ \begin{array}{ll} \mathrm{f}(\mathrm{t})=\mathrm{t}^{2}, & 0<\mathrm{t}<2 \\ \text { and }=6, & \mathrm{t}>2 . \end{array} $$

Let \(\mathrm{f}\) be a periodic function with period \(\mathrm{T}\), i.e., $$ f(t+T)=f(t) $$ for all \(t\) (see Figure), and let $$ \begin{array}{ll} g_{i}(t)=f(t) & (0 \leq t \leq T) \\ \text { and }=0 & (t>T) \end{array} $$ Show that $$ F(s)=\left[\\{G(s)\\} /\left(1-e^{-s T}\right)\right] $$ where \(\mathrm{F}\) and \(\mathrm{G}\) are the Laplace transforms of \(\mathrm{f}\) and \(\mathrm{g}_{1}\) respectively.

Prove that if $$ f(x+b)=-f(x) $$ for all \(\mathrm{x}\), where \(\mathrm{b}\) is a constant, then $$ L\\{f(t)\\}=\left[\left\\{b \int_{0} e^{-s t} f(t) d t\right\\} /\left(1+e^{-b s}\right)\right] $$ where \(L\) is the Laplace transform operator. Functions satisfying (1) are often called antiperiodic and are very important in electrical engineering.

Find the Laplace transform \(L\\{h(t)\\}\), where $$ \begin{array}{ll} \mathrm{h}(\mathrm{t})=1, & 0<\mathrm{t}<\mathrm{c} \\ \text { and }=-1, & \mathrm{c}<\mathrm{t}<2 \mathrm{c} \end{array} $$ and \(\mathrm{h}(\mathrm{t}+2 \mathrm{c})=\mathrm{h}(\mathrm{t})\) for all \(\mathrm{t}\), with c a constant in the following two ways. (See Figure 1.) (a) Use the fact that $$ L\\{g(t)\\}=\left[1 /\left\\{s\left(1+e^{-c s}\right)\right\\}\right] $$ where $$ \begin{array}{ll} g(t)=1, & 0

Find the inverse Laplace transforms (a) \(L^{-1}\left[2 \mathrm{~s} /\left(\mathrm{s}^{2}+1\right)^{2}\right]\) (b) \(L^{-1}[1 / \sqrt{s}]\)

State a theorem which gives conditions under which two functions must be identical if they have the same Laplace transform, i.e., within what class of functions is the inverse Laplace transform unique? What does this imply about the functions $$ \mathrm{f}(\mathrm{t})=\mathrm{L}^{-1}\left\\{\mathrm{aF}_{1}(\mathrm{~s})+\mathrm{bF}_{2}(\mathrm{~s})\right\\} $$ and $$ \underline{f}(t)=a L^{-1}\left\\{F_{1}(s)\right\\}+b L^{-1}\left\\{F_{2}(s)\right\\} $$ where \(\mathrm{F}_{1}(\mathrm{~s})\) and \(\mathrm{F}_{2}(\mathrm{~s})\) are some Laplace transforms?

Find and sketch the function \(g(t)\) which is the inverse Laplace transform $$ g(t)=L^{-1}\left[(3 / s)-\left(4 e^{-s} / s^{2}\right)+\left(4 e^{-3 s} / s^{2}\right)\right] $$

Find the inverse Laplace transform $$ \mathrm{f}(\mathrm{t})=\mathrm{L}^{-1}[\log \\{(\mathrm{s}+1) /(\mathrm{s}-1)\\}], \quad \mathrm{s}>1 $$

Find the inverse Laplace transforms (a) \(L^{-1}\left[1 /\left(s^{2}-2 s+9\right)\right]\), (b) \(L^{-1}\left[(s+1) /\left(s^{2}+6 s+25\right)\right]\)

Outline the general method of rational fraction decomposition used to find the inverse Laplace transform of a rational function $$ F(s)=P(s) / Q(s) $$ where \(\mathrm{Q}(\mathrm{s})\) is of higher degree than \(\mathrm{P}(\mathrm{s})\).

Find the inverse Laplace transforms $$ \begin{aligned} \mathrm{f}(\mathrm{t}) &=\mathrm{L}^{-1}\\{\mathrm{~F}(\mathrm{~s})\\} \\ &=\mathrm{L}^{-1}\left[\left(3 \mathrm{~s}^{2}+17 \mathrm{~s}+47\right) /\left\\{(\mathrm{s}+2)\left(\mathrm{s}^{2}+4 \mathrm{~s}+29\right)\right\\}\right] \end{aligned} $$

Use partial fractions to decompose (a) \(\left[1 /\left\\{(\mathrm{s}+1)\left(\mathrm{s}^{2}+1\right)\right\\}\right]\) (b) \(\left[1 /\left\\{\left(s^{2}+1\right)\left(s^{2}+4 s+8\right)\right\\}\right]\).

(a) Define the convolution of two functions \(\mathrm{f}(\mathrm{t})\) and \(\mathrm{g}(\mathrm{t})\). (b) State the convolution theorem for Laplace transforms. (c) Find the inverse Laplace transform $$ \mathrm{f}(\mathrm{t})=\mathrm{L}^{-1}\\{\mathrm{~F}(\mathrm{~S})\\}=\mathrm{L}^{-1}\left\\{1 /\left(\mathrm{s}^{2}+\mathrm{c}^{2}\right)^{2}\right\\} $$ \((\mathrm{c}=\) constant \() .\)

Find the solution to the initial value problem $$ \mathrm{y}^{\prime \prime}(\mathrm{t})+4 \mathrm{y}^{\prime}(\mathrm{t})+8 \mathrm{y}(\mathrm{t})=\sin \mathrm{t} $$ where $$ \mathrm{y}(0)=1 \quad \text { and } \quad \mathrm{y}^{\prime}(0)=0 $$

Solve the initial value problem $$ \begin{aligned} &y^{\prime \prime}(t)+2 y^{\prime}(t)+5 y(t)=H(t) \\ &y(0)=y^{\prime}(0)=0 \end{aligned} $$ where $$ \mathrm{H}(\mathrm{t})=1, \quad 0 \leq \mathrm{t}<\pi $$ $$ \text { and }=0, \quad t \geq \pi, $$ as shown in the accompanying graph.