MATH 308 Lecture 20

« previous | Monday, March 4, 2013 | next »

Lecture Notes

Section 6.2

Exercise 11

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'' + 3t\,y' - 6y = 1 \quad y(0)=0 \quad y'(0) = 0}

No way to solve this... until now.

${\begin{aligned}{\mathcal {L}}\left\{y''\right\}+3{\mathcal {L}}\left\{t\,y'\right\}-6{\mathcal {L}}\left\{y\right\}&={\mathcal {L}}\left\{1\right\}\\s^{2}\,{\mathcal {L}}\left\{y\right\}-y'(0)-s\,y(0)+3\left(-{\frac {\mathrm {d} }{\mathrm {d} s}}\left({\mathcal {L}}\left\{y'\right\}\right)\right)-6{\mathcal {L}}\left\{y\right\}&={\frac {1}{s}}\\\left(s^{2}-9\right)\,{\mathcal {L}}\left\{y\right\}-3s\,{\frac {\mathrm {d} }{\mathrm {d} s}}\,{\mathcal {L}}\left\{y\right\}&={\frac {1}{s}}\end{aligned}}$

Notice that we have a differential equation: let $Y(s)={\mathcal {L}}\left\{y\right\}(s)$

${\begin{aligned}-3s\,Y'(s)+\left(s^{2}-9\right)\,Y(s)&={\frac {1}{s}}\\Y'(s)-{\frac {s^{2}-9}{3s}}\,Y(s)&=-{\frac {1}{3s^{2}}}\\\mu (s)&=\mathrm {e} ^{-\int {\frac {s^{2}-9}{3s}}\,\mathrm {d} s}\\&=s^{3}\,\mathrm {e} ^{-{\frac {s^{2}}{6}}}\\{\frac {\mathrm {d} }{\mathrm {d} s}}\left(s^{3}\,\mathrm {e} ^{-{\frac {s^{2}}{6}}}\,Y(s)\right)&=-{\frac {1}{3}}\,s\,\mathrm {e} ^{-{\frac {s^{2}}{6}}}\\s^{3}\,\mathrm {e} ^{-{\frac {s^{2}}{6}}}\,Y(s)&=\mathrm {e} ^{-{\frac {s^{2}}{6}}}+C\\Y(s)&={\frac {\mathrm {e} ^{-{\frac {s^{2}}{6}}}}{s^{3}\,\mathrm {e} ^{-{\frac {s^{2}}{6}}}}}+{\frac {C\,\mathrm {e} ^{\frac {s^{2}}{6}}}{s^{3}}}\end{aligned}}$

$C$ must be 0 since ${\mathcal {L}}\left\{y\right\}$ must converge to 0 when $s$ approaches infinity.

${\begin{aligned}{\mathcal {L}}\left\{y\right\}=Y(s)&={\frac {1}{s^{3}}}\\&={\frac {1}{2}}\,{\mathcal {L}}\left\{t^{2}\right\}\\y&={\frac {t^{2}}{2}}\end{aligned}}$

This solution satisfies the initial conditions and the differential equation.

Section 6.3

Lecture Notes

Heaviside Function

The unit step function or Heaviside function is the function $u_{c}$ defined by

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} u_c(t) &= \begin{cases} 0 & t < c \\ 1 & t \ge c \end{cases} & c \ge 0 \end{align}}

Application: string suddenly breaks in a physics problem.

Exercise 1

Sketch graph of 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(t) = u_2(t) + 3u_4(t) - 7 u_5(t)}

in Maple:

> restart: with(plots): with(plottools):
> u := (s,t) -> piecewise(t<s, 0, t>=s, 1):
> plot(u(2,t) + 3*u(4,t) - 7*u(5,t), t=0..10, discont=true);

Sketch graph of 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(t) = (t-1) \, u_2(t) + 2(t-3) \, u_3(t)}

Exercise 2

Express the following functions in terms of step 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(t) = \begin{cases} t & 0 \le t < 3 \\ 1 - t & 3 \le t < 5 \\ t^2 & 5 \le t < 6 \\ 3 & t \ge 6 \end{cases}}

Something happens at 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 t = \{ 3,5,6 \}} , so we can use 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 u_c(t)} to "turn on" different parts of the function at each "action" point.

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) = A + B \, u_3(t) + C \, u_5(t) + D \, u_6(t)}

• Between 0 and 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 f(t) = t} , so 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 A = t}
• Between 3 and 5, 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) = 1 - t} , so 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 B = 1-2t}
• Between 5 and 6, 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) = t^2} , so 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 = t^2 + t - 1}
• After 6, 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) = 3} , so 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 D = 3 - t^2}

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) = t + (1-2t) \, u_3(t) + (t^2+t-1) \, u_5(t) + (3-t^2) \, u_6(t)}

Theorem 6.3.1

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(s) = \mathcal{L} \left\{ f(t) \right\}} exists for 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 s > a \ge 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 positive 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\{ u_c(t) \, f(t-c) \right\} = \mathrm{e}^{-c \, s} \, \mathcal{L} \left\{ f(t) \right\} = \mathrm{e}^{-c \, s} \, F(s) \quad s > a}

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 u_c(t) \, f(t-c) = \mathcal{L}^{-1} \left\{ \mathrm{e}^{-c\,s}\, F(s) \right\}}

Exercise 3

Find the Laplace transform of the function

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} f(t) &= \begin{cases} 0 & t < \pi \\ t-\pi & pi \le t \le 2\pi \\ 0 & t \ge 2\pi \end{cases} \\ &= (t-\pi) \, u_\pi(t) + (\pi-t) \, u_{2\pi}(t) \\ \mathcal{L}\left\{ f(t) \right\} &= \mathcal{L} \left\{ (t-\pi) \, u_\pi (t) \right\} + \mathcal{L} \left\{ (\pi-t) \, u_{2\pi} \right\} \\ \mathcal{L} \left\{ (t-\pi) \, u_\pi(t) \right\} &= \mathrm{e}^{-\pi \, s} \, \mathcal{L} \left\{ g(t) \right\} \quad \quad g(t) = t \quad g(t-\pi) = t - \pi \\ \mathcal{L} \left\{ (\pi-t) \, u_{2\pi} \right\} &= \mathrm{e}^{-2\pi \, s} \, \mathcal{L} \left\{ h(t) \right\} \\ h(t-2\pi) &= \pi-t \\ &= \pi - (t - 2\pi + 2\pi) \\ &= -(t-2\pi)-\pi \\ h(T) &= -T-\pi \\ \mathcal{L} \left\{ f(t) \right\} &= \mathrm{e}^{-\pi \, s} \, \mathcal{L} \left\{ t \right\} + \mathrm{e}^{-2\pi \, s} \, \mathcal{L} \left\{ -t-\pi \right\} \end{align}}