Mth3006 Weekly Problems 3

Question One

Using the definition of a Laplace transform, $\tilde{f}(s)={\int }_0^{\infty }f(t)e^{-st}dt$, calculate the Laplace transforms of the following functions, where $a\ne 0$ is a real constant:

1. $a(x)=1$,
2. $b(x)=e^{at}$ (and state the range for of $s$ for which the transform exists),
3. $c(x)=\cosh (at)$ (and state the range for of $s$ for which the transform exists),
4. $d(x)=\sinh (at)$ (and state the range for of $s$ for which the transform exists).

Solution

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Question Two

Use the result that $\mathcal{L}[t^{n}f(t)]=(-1{)}^n\frac{d^n}{d{s}^n}\tilde{f}(s)$, as well as the Table of Laplace transforms, to evaluate the following Laplace transforms:

1. $\mathcal{L}[t{e}^{-2t}]$,
2. $\mathcal{L}[t^2{e}^{-2t}]$.

Solution

Part one can be completed by…

$$\begin{array}{rl}\mathcal{L}[t{e}^{-2t}]& =(-1)\frac{d}{ds}\tilde{f}(s)\\ & =-\frac{d}{ds}\left(\frac{1}{s+2}\right)\\ & =-\frac{d}{ds}(s+2{)}^{-1}\\ & =-(s+2{)}^{-2}\\ & =\boxed{-\frac{1}{(s+2{)}^2}}\end{array}$$

Whilst part evaluates to…

$$\begin{array}{rl}\mathcal{L}[t^2{e}^{-2t}]& =(-1{)}^2\frac{d^2}{d{s}^2}\tilde{f}(s)\\ & =\frac{d^2}{d{s}^2}(s+2{)}^{-1}\\ & =(s+2{)}^{-3}\\ & =\boxed{\frac{1}{(s+2{)}^3}}\end{array}$$

Showing that the power of $t$ just dictates the number of times you differentiate.

Question Three

If $\tilde{F}(s)=\frac{a}{s^2(s^2+a^2)}$, where $a>0$ is a constant, calculate $f(t)={\mathcal{L}}^{-1}[\tilde{F}(s)]$ using:

1. The convolution theorem,
2. Partial fractions.

Solution

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Question Four

Use Laplace transforms to show that…

$${\int }_0^{\infty }{e}^{-2t}(1-\cos 3t)dt=\frac{9}{26}$$

Solution

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Question Five

Use the convolution theorem to show the following, where $a$ is a real, non-zero constant:

$${\mathcal{L}}^{-1}\left[\frac{s}{s^2+a^2}\right]=\frac{t\sin (at)}{2a}$$

Solution

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