MATH 308 Lecture 21

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Lecture Notes

Theorem 6.3.1

If $F(s)={\mathcal {L}}\left\{f(t)\right\}$ exists for $s>a\geq 0$ and if $c$ is a positive constant, then

${\mathcal {L}}\left\{u_{c}(t)h(t-c)\right\}=\mathrm {e} ^{-c\,s}\,{\mathcal {L}}\left\{f(t)\right\}=\mathrm {e} ^{-c\,s}\,F(s)\quad s>a$

Exercise 4

Find the Laplace transform of $f(t)={\begin{cases}2t&0\leq t<2\\t^{2}&2\leq t<5\\0&5\leq t\end{cases}}$

Rewrite $f(t)=2t+u_{2}(t)\left(t^{2}-2t\right)+u_{5}(t)\left(-t^{2}\right)$

${\mathcal {L}}\left\{f(t)\right\}=2{\mathcal {L}}\left\{t\right\}+{\mathcal {L}}\left\{u_{2}(t)\,(t^{2}-2t)\right\}-{\mathcal {L}}\left\{u_{5}(t)\,t^{2}\right\}$

For second term, let $h(t-2)=t^{2}-2t$, then $h(t)=t^{2}+2t$

For third term, let $h(t-5)=t^{2}$, then $h(t)=x^{2}+10x+25$

Then we have

${\mathcal {L}}\left\{f(t)\right\}={\frac {2}{s^{2}}}+\mathrm {e} ^{-2s}\left({\frac {2}{s^{3}}}+{\frac {2}{s^{2}}}\right)-\mathrm {e} ^{-5s}\left({\frac {2}{s^{3}}}+{\frac {10}{s^{2}}}-{\frac {25}{s}}\right)$

Theorem 6.3.2

If $F(s)={\mathcal {L}}\left\{f(t)\right\}$ exists for $s>a\geq 0$ and if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is a constant, then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ \mathrm{e}^{c\,t} \, f(t) \right\} = F(s-c) \quad s > a+c}

Conversely, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) = \mathcal{L}^{-1} \left\{ F(s) \right\}} , then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^{c\,t} \, f(t) = \mathcal{L}^{-1} \left\{ F(s-c) \right\}}

Exercise 5

Find the inverse Laplace transform of the functions:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(s) = \frac{\mathrm{e}^{-2x}}{s^2-2s-3}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(s) = \frac{\mathrm{e}^{-s}-\mathrm{e}^{-3s}+3}{s}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(s) = \frac{2 \mathrm{e}^{-3s}}{(s-1)^2 + 4}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathcal{L}^{-1} \left\{ G(s) \right\} &= \mathcal{L}^{-1} \left\{ \frac{\mathrm{e}^{-s}}{s} - \frac{\mathrm{e}^{-3s}}{s} + \frac{3}{s} \right\} \\ &= u_1(t) - u_3(t) + 3 \end{align}}

Exercise 7

Find solution of initial value problems Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y''+y = u_{3\pi}(t)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0) = 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(0) = 1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathcal{L} \left\{ y'' \right\} + \mathcal{L} \left\{ y \right\} &= \mathcal{L} \left\{ 1 - u_{3\pi}(t) \right\} \\ s^2 \mathcal{L} \left\{ y \right\} - y'(0) - s \, y(0) + \mathcal{L} \left\{ y \right\} &= \frac{1-\mathrm{e}^{-3\pi\,s}}{s} \\ (s^2 + 1) \mathcal{L} \left\{ y \right\} - 1 &= \frac{1-\mathrm{e}^{-3\pi\,s}}{s} \end{align}}