MATH 308 Lecture 19

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Lecture Notes

Section 6.1–6.2

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

Partial Fraction Decomposition from last time.

${\begin{aligned}{\mathcal {L}}\left\{y\right\}&={\frac {s+1}{s(s-2)(s+2)}}\\&={\frac {a}{s}}+{\frac {b}{s-2}}+{\frac {c}{s+2}}\end{aligned}}$

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

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

Therefore,

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

Inverse Laplace Transform (Exercise 7)

${\begin{aligned}{\mathcal {L}}\left\{y\right\}&={\frac {3s^{2}+s}{(s+1)(s-1)^{2}}}\\&={\frac {a}{s+1}}+{\frac {b}{(s-1)^{1}}}+{\frac {c}{(s-1)^{2}}}\end{aligned}}$

1. Multiply by $s+1$ and take $s=-1$
2. Multiply by $(s-1)^{2}$ and take $s=1$

${\begin{aligned}{\frac {3s^{2}+s}{(s-1)^{2}}}&=a+{\frac {b(s+1)}{(s-1)^{1}}}+{\frac {c(s+1)}{(s-1)^{2}}}\\{\frac {1}{2}}&=a&{\text{when}}\ s&=-1\\{\frac {3s^{2}+s}{s+1}}&={\frac {a(s-1)^{2}}{s+1}}+b(s-1)+c\\2&=c&{\text{when}}\ s&=1\end{aligned}}$

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}} )