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Problem 468

(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 \(\alpha>0\)

Expert verified

We proved that the functions f(t) = t^n, n>0, and f(t) = sin(kt) are of exponential order α on [0, ∞] for all α>0. For f(t) = t^n, we used L'Hôpital's Rule to show that the limit t^n/e^(αt) goes to 0 as t approaches infinity. For f(t) = sin(kt), we showed that |sin(kt)| ≤ e^(αt) holds true for all α>0 since e^(αt) is non-negative and |sin(kt)| is always less than or equal to 1.

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Chapter 16

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.

Chapter 16

(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 \() .\)

Chapter 16

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

Chapter 16

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 $$ \mathrm{F}(\mathrm{s})=\mathrm{L}\\{\mathrm{f}(\mathrm{t})\\} $$

Chapter 16

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.

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