MATH 308 Lecture 20
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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.
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\} + 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{align}}
Notice that we have a differential equation: let 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(s) = \mathcal{L} \left\{ y \right\} (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 \begin{align} -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{align}}
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} must be 0 since 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\{ y \right\}} must converge to 0 when 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} approaches infinity.
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\} = Y(s) &= \frac{1}{s^3} \\ &= \frac{1}{2} \, \mathcal{L} \left\{ t^2 \right\} \\ y &= \frac{t^2}{2} \end{align}}
This solution satisfies the initial conditions and the differential equation.
Section 6.3
Heaviside Function
The unit step function or Heaviside function is 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 u_c} defined by
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
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
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}}