MATH 308 Lecture 37

« previous | Friday, April 26, 2013 | next »

Lecture Notes

Section 7.9

Exercise 1c: Imaginary Eigenvectors

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 X'(t) = \begin{bmatrix}2&-5\\1&-2\end{bmatrix} \, X + \begin{bmatrix} -\cos{t} \\ \sin{t} \end{bmatrix}}

Eigenvalues: 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 \lambda \in \{ -i, i \}}

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 X_h(t) = c_1 \, \left( \vec{v}_1 \, \cos{t} - \vec{v}_2 \, \sin{t} \right) + c_2 \, \left( \vec{v}_2 \, \cos{t} + \vec{v}_1 \, \sin{t} \right)}

Guess 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 X_p(t)} :

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 X-p(t) = t \, \vec{a} \, \cos{t} + t \, \vec{b} \, \sin{t} + \vec{c} \, \cos{t} + \vec{d} \sin{t}}

Exercise 2a: Laplace Transform

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 X'(t) = \begin{bmatrix}0&1\\1&0\end{bmatrix} \, X + \begin{bmatrix} t \\ -1 \end{bmatrix} \quad \quad X(0) = \begin{bmatrix}2\\1\end{bmatrix}}

${\begin{cases}{\mathcal {L}}\left\{x'\right\}={\mathcal {L}}\left\{y\right\}+{\mathcal {L}}\left\{t\right\}\\{\mathcal {L}}\left\{y'\right\}={\mathcal {L}}\left\{x\right\}+{\mathcal {L}}\left\{-1\right\}\end{cases}}$

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{cases} s \, \mathcal{L} \left\{ x \right\} - x(0) = \mathcal{L} \left\{ y \right\} + \frac{1}{s^2} \\ s \, \mathcal{L} \left\{ y \right\} - y(0) = \mathcal{L} \left\{ x \right\} + \frac{1}{s} \end{cases} }

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{cases} s \, \mathcal{L} \left\{ x \right\} - 2 = \mathcal{L} \left\{ y \right\} + \frac{1}{s^2} \\ s \, \mathcal{L} \left\{ y \right\} - 1 = \mathcal{L} \left\{ x \right\} + \frac{1}{s} \end{cases} }

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\{ x \right\} = s \, \mathcal{L} \left\{ y \right\} - 1 + \frac{1}{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 \mathcal{L} \left\{ y \right\} = \frac{1}{s^2 \, \left( s^2 - 1 \right)} + \frac{1}{s^2-1} + \frac{s}{s^2-1}}

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 \mathcal{L} \left\{ x \right\} = s \, \left( \frac{1}{s^2 \, \left( s^2 - 1 \right)} + \frac{1}{s^2-1} + \frac{s}{s^2-1} \right) - 1 + \frac{1}{s}}

Take inverse Laplace transforms

Exercise 2b

Using matrix notation,

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 X'(t) = \begin{bmatrix}2 & -1 \\ 3 & -2 \end{bmatrix} \, X(t) + \begin{bmatrix} 1 \\ -1 \end{bmatrix} \, \mathrm{e}^{t} \quad \quad X(0) = \begin{bmatrix}3 \\ 0 \end{bmatrix}}

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\{ X' \right\} &= \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \, \mathcal{L} \left\{ X \right\} + \begin{bmatrix}1,-1\end{bmatrix} \, \mathcal{L} \left\{ \mathrm{e}^{t} \right\} \\ s \, \mathcal{L} \left\{ X \right\} - X(0) &= \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \, \mathcal{L} \left\{ X \right\} + \begin{bmatrix}1 \\ -1 \end{bmatrix} \, \left( \frac{1}{s-1} \right) \\ \left( s\,I - \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \right) \, \mathcal{L} \left\{ X \right\} &= X(0) + \frac{1}{s-1} \, \begin{bmatrix}1\\-1\end{bmatrix} \\ \mathcal{L} \left\{ X \right\} &= \begin{bmatrix}s-2&1\\-3&s+2\end{bmatrix}^{-1} \, \left( X(0) + \frac{1}{s-1} \, \begin{bmatrix}1\\-1\end{bmatrix} \right) \\ \mathcal{L} \left\{ X \right\} &= \frac{1}{s^2-1} \, \begin{bmatrix}s+2 & -1 \\ 3 & s-2 \end{bmatrix} \, \begin{bmatrix} 3 + \frac{1}{s-1} \\ -\frac{1}{s-1} \end{bmatrix} \end{align}}