Simon's copy of Laplace of constants and powers of t

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Question 1

Use the following standard Laplace transform results in the questions below:

$L[af(t)+bg(t)]=aL[f(t)]+bL[g(t)]$ (Linearity)

$L[k]=\frac{k}{s}$

$L[t]=\frac{1}{s^2}$

$L[t^n]=\frac{n!}{s^{n+1}}$

a)

Find $L[\var{a} ]$ .

$\displaystyle{}$ Expected answer:

$\displaystyle{\frac{ 5 }{ s }}$

Score: 0/1

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b)

Find $L[\var{b}t ]$ .

$\displaystyle{}$ Expected answer:

$\displaystyle{\frac{ 8 }{ s^{ 2 } }}$

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c)

Find $L[\var{a}+\var{b}t ]$ .

$\displaystyle{}$ Expected answer:

$\displaystyle{\frac{ 5 }{ s } + \frac{ 8 }{ s^{ 2 } }}$

Score: 0/1

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d)

Find $L[t^\var{a}]$ .

$\displaystyle{}$ Expected answer:

$\displaystyle{\frac{ 5! }{ s^{ 6 } }}$

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e)

Find $L[\var{a}t^\var{c}+\var{b}t^\var{d}]$ .

$\displaystyle{}$ Expected answer:

$\displaystyle{\frac{ 5 \times 2! }{ s^{ 3 } } + \frac{ 8 \times 9! }{ s^{ 10 } }}$

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Advice

(a)

$L[k]=\frac{k}{s}$

Hence $L[\var{a} ]=\frac{\var{a}}{s}$ .

(b)

Note $L[\var{b}t ]=\var{b}L[t]$ by linearity

Now we use $L[t]=\frac{1}{s^2}$ to obtain:

$L[\var{b}t ]=\var{b}L[t]=\var{b}\frac{1}{s^2} =\frac{\var{b}}{s^2}$

(c)

Use linearity to obtain

$L[\var{a}+\var{b}t ]=L[\var{a}]+L[\var{b}t ]=\frac{\var{a}}{s}+\frac{\var{b}}{s^2}$ using our results from parts (a) and (b)

(d)

We use $L[t^n]=\frac{n!}{s^{n+1}}$

Hence $L[t^\var{a}]=\frac{\var{a}!}{s^{\var{a}+1}}=\simplify{{a}!/(s^({a}+1))}$ .

(e)

By linearity, $L[\var{a}t^\var{c}+\var{b}t^\var{d}]=\var{a}L[t^\var{c}]+\var{b}L[t^\var{d}]$ .

Now we use $L[t^n]=\frac{n!}{s^{n+1}}$ to obtain

$L[\var{a}t^\var{c}+\var{b}t^\var{d}]=\var{a}L[t^\var{c}]+\var{b}L[t^\var{d}]= \var{a}\times \frac{\var{c}!}{s^{\var{c}+1}} +\var{b} \times \frac{\var{d}!}{s^{\var{d}+1}}=\frac{\var{a}\times\var{c}!}{s^{\var{c+1}}}+\frac{\var{b} \times\var{d}!}{s^{\var{d+1}}}$

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Numbas has failed

Simon's copy of Laplace of constants and powers of t

a)

b)

c)

d)

e)

Advice