MATH 308 Lecture 19

From Notes
Jump to navigation Jump to search

« previous | Friday, March 1, 2013 | next »

Lecture Notes


Section 6.1–6.2

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 y'' - 4y = 1 \quad y(0) = 0 \quad y'(0) = 1}

Partial Fraction Decomposition from last time.

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\} &= \frac{s+1}{s(s-2)(s+2)} \\ &= \frac{a}{s}+\frac{b}{s-2}+\frac{c}{s+2} \end{align}}

Multiply by each term 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 q} in the denominator and take 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 q = 0}

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} \frac{s+1}{(s-2)(s+2} &= a + \frac{bs}{s-2} + \frac{cs}{s+2} \\ -\frac{1}{4} &= a & \text{when}\ s &= 0 \\ \frac{s+1}{s(s+2)} &= \frac{a(s-2)}{s} + b + \frac{c(s-2)}{s+2} \\ \frac{3}{8} &= b & \text{when}\ s &= 2 \\ \frac{s+1}{s(s-2)} &= \frac{a(s+2)}{s} + \frac{b(s+2)}{s-2} + c \\ -\frac{1}{8} &= c & \text{when}\ s &= -2 \end{align}}

Therefore,

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\} &= -\frac{1}{4} \, \frac{1}{s} + \frac{3}{8} \, \frac{1}{s-2} - \frac{1}{8} \, \frac{1}{s+2} \\ &= -\frac{1}{4} \, \mathcal{L} \left\{ 1 \right\} + \frac{3}{8} \, \mathcal{L} \left\{ \mathrm{e}^{2t} \right\} - \frac{1}{8} \, \mathcal{L} \left\{ \mathrm{e}^{-2t} \right\} \\ &= \mathcal{L} \left\{ -\frac{1}{4} + \frac{3}{8} \, \mathrm{e}^{2t} - \frac{1}{8} \, \mathrm{e}^{-2t} \right\} \\ y &= -\frac{1}{4} + \frac{3}{8} \, \mathrm{e}^{2t} - \frac{1}{8} \, \mathrm{e}^{-2t} \end{align}}

Inverse Laplace Transform (Exercise 7)

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\} &= \frac{3s^2+s}{(s+1)(s-1)^2} \\ &= \frac{a}{s+1} + \frac{b}{(s-1)^1} + \frac{c}{(s-1)^2} \end{align}}


  1. Multiply 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 s+1} and take 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 = -1}
  2. Multiply 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 (s-1)^2} and take 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 = 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} \frac{3s^2+s}{(s-1)^2} &= a + \frac{b(s+1)}{(s-1)^1} + \frac{c(s+1)}{(s-1)^2} \\ \frac{1}{2} &= a & \text{when}\ s &= -1 \\ \frac{3s^2+s}{s+1} &= \frac{a(s-1)^2}{s+1} + b(s-1) + c \\ 2 &= c & \text{when}\ s &= 1 \end{align}}


Taking similar steps won't work 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 b} since taking 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 = 1} will cause a division by zero on the "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} term".

Multiply 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 s} , take 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 \lim_{s \to \infty}} , and take 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 = 0}

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} \lim_{s\to\infty} \frac{3s^3}{s^3} &= 3 = a + b \\ 3 &= \frac{1}{2} + b \\ b &= \frac{5}{2} \end{align}}


(I'm not sure what's going on here...)

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} 0 &= \frac{1}{2} - b + 2 \\ b &= \frac{1}{2} + 2 \end{align}}


Therefore

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\} &= \frac{1}{2} \, \frac{1}{s+1} + \frac{5}{2} \, \frac{1}{s-1} + \frac{2}{(s-1)^2} \\ &= \frac{1}{2} \, \mathcal{L} \left\{ \mathrm{e}^{-t} \right\} + \frac{5}{2} \, \mathcal{L} \left\{ \mathrm{e}^{t} \right\} + 2 \mathcal{L} \left\{ t\,\mathrm{e}^{t} \right\} \\ y &= \frac{\mathrm{e}^{-t}}{2} + \frac{5 \mathrm{e}^{t}}{2} + 2t\,\mathrm{e}^{t} \end{align}}

Exercise 8

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} 12 \mathrm{e}^t &= y'' + 6y' + 5y \quad\quad y(0) = -1 \quad y'(0) = 7 \\ 12 \mathcal{L} \left\{ \mathrm{e}^t \right\} &= \mathcal{L} \left\{ y'' \right\} + 6\mathcal{L} \left\{ y' \right\} + 5 \mathcal{L} \left\{ y \right\} \\ \frac{12}{s-1} &= s^2 \, \mathcal{L} \left\{ y \right\} - y'(0) -5y(0) + 6s\, \mathcal{L} \left\{ y \right\} - 6y(0) + 5 \mathcal{L} \left\{ y \right\} \\ \frac{11-s^2+2s}{s-1} &= \left( s^2 + 6s + 5 \right) \, \mathcal{L} \left\{ y \right\} \\ \mathcal{L} \left\{ y \right\} &= \frac{11-s^2+2s}{(s-1)(s+5)(s+1)} = \frac{a}{s-1} + \frac{b}{s+5} + \frac{c}{s+1} \\ &= \frac{1}{s-1} - \frac{1}{s+5} - \frac{1}{s+1} \\ &= \mathcal{L} \left\{ \mathrm{e}^{t} \right\} - \mathcal{L} \left\{ \mathrm{e}^{-5t} \right\} - \mathcal{L} \left\{ \mathrm{e}^{-t} \right\} \\ &= \mathcal{L} \left\{ \mathrm{e}^{t} - \mathrm{e}^{-5t} - \mathrm{e}^{-t} \right\} \\ y &= \mathrm{e}^{t} - \mathrm{e}^{-5t} - \mathrm{e}^{-t} \end{align}}

Exercise 9

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 w'' + w = t^2 +2 \quad w(0) = 1 \quad w'(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\{ w'' \right\} + \mathcal{L} \left\{ w \right\} &= \mathcal{L} \left\{ t^2 + 2 \right\} \\ s^2 \mathcal{L} \left\{ w \right\} - w'(0) - s \, w(0) + \mathcal{L}\left\{ w \right\} &= \frac{2}{s^3} + \frac{2}{5} \\ \mathcal{L} &= \frac{2 + 2s^2 - s^3 + s^4}{s^3(s^2+1)} = \frac{a}{s} + \frac{b}{s^2} + \frac{c}{s^3} + \frac{ds+e}{s^2+1} \end{align}}

We find 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 = 2} easily, but lower degrees are harder: multiply 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 s} and take limit to 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 1 = a + d}

(out of time; WolframAlpha says the decomposition is 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 \frac{2}{s^3} + \frac{s-1}{s^2+1}} )

Retrieved from "https://notes.komputerwiz.net/w/index.php?title=MATH_308_Lecture_19&oldid=2894"