MATH 308 Lecture 39
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Review
Exercise 24.1
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 X' = \begin{bmatrix}1&1\\4&-2\end{bmatrix} \, X + \begin{bmatrix}\mathrm{e}^{-2t}\\-2 \mathrm{e}^{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 \left\{ -3, 2 \right\}}
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 \vec{x}_1 = \left\langle 1,-4 \right\rangle \quad \vec{x}_2 = \left\langle 1,1 \right\rangle}
Geneal Solution to Homogeneous Problem: 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_1 \, \mathrm{e}^{-3t} \, \begin{bmatrix}1,-4\end{bmatrix} + c_2 \, \mathrm{e}^{2t} \, \begin{bmatrix}1\\1\end{bmatrix}}
Undetermined Coefficients
Guess for particular solution: 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) = \vec{v}_1 \, \mathrm{e}^{-2t} + \vec{v}_2 \, \mathrm{e}^{t}}
Derivation and Solution:
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} X'(t) &= -2 \mathrm{e}^{-2t} \, \vec{v}_1 + \mathrm{e}^{t} \vec{v}_2 \\ A \, X(t) + G(t) &= \begin{bmatrix}1&1\\4&-2\end{bmatrix} \, \vec{v}_1 + \mathrm{e}^{t} \, \begin{bmatrix}1&1\\4&-2\end{bmatrix} \, \vec{v}_2 + \begin{bmatrix}\mathrm{e}^{-2t} \\ -2 \mathrm{e}^{t} \end{bmatrix} \\ -2 \mathrm{e}^{-2t} \, \vec{v}_1 + \mathrm{e}^{t} \, \vec{v}_2 &= \mathrm{e}^{-2t} \, \begin{bmatrix}a+b\\a-2b\end{bmatrix} + \mathrm{e}^{t} \, \begin{bmatrix}c+d \\ 4c-2d\end{bmatrix} + \mathrm{e}^{-2t} \begin{bmatrix}1\\0\end{bmatrix} + \mathrm{e}^{t} \, \begin{bmatrix} 0\\-2 \end{bmatrix} \\ \end{align}}
Coefficient 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 \mathrm{e}^{2t}} :
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} -2a = a+b+1 -2b = 4a-2b \end{cases}}
Coefficient 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 \mathrm{e}^{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 \begin{cases} c = c+d d = 4c-2d \end{cases}}
Variation of Parameters
Recall 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_1' \, X_1 + c_2' \, X_2 = G(t)}
Plugging in our values gives:
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_1' \, \mathrm{e}^{-3t} \, \begin{bmatrix}1\\-4\end{bmatrix} + c_2' \, \mathrm{e}^{2t} \, \begin{bmatrix}1\\1\end{bmatrix} = \begin{bmatrix}\mathrm{e}^{-2t} \\ -2 \mathrm{e}^{t}\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{cases} c_1' \, \mathrm{e}^{-3t} + c_2' \, \mathrm{e}^{2t} = \mathrm{e}^{-2t} -4c_1' \, \mathrm{e}^{-3t} + c_2' \, \mathrm{e}^{2t} = -2 \mathrm{e}^{t} \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{align} c_1' &= \frac{\mathrm{e}^{t} + 2 \mathrm{e}^{4t}}{5} \\ c_2' &= \left( \mathrm{e}^{-2t} - \mathrm{e}^{3t} \, \left( \frac{\mathrm{e}^{t} + 2 \mathrm{e}^{4t}}{5} \right) \right) \mathrm{e}^{-2t} \end{align}}
Take the antiderivatives of each function to get the final constants.
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 \begin{align} \mathcal{L} \left\{ X'(t) \right\} &= A \, \mathcal{L} \left\{ X(t) \right\} + \mathcal{L} \left\{ G(t) \right\} \\ s \, \mathcal{L} \left\{ X(t) \right\} - X(0) &= A \, \mathcal{L} \left\{ X(t) \right\} + \begin{bmatrix} \frac{1}{s+2} \\ -\frac{2}{s-1} \end{bmatrix} \end{align}}
Let the initial condition 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(0) = \left\langle 0,0 \right\rangle} (we can pick whatever we want)
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} \left( s \, I - A \right) \, \mathcal{L} \left\{ X(t) \right\} &= \begin{bmatrix} \frac{1}{s-2} \\ -\frac{2}{s-1} \end{bmatrix} \\ \begin{bmatrix} s-1 & -1 \\-4 & s+2\end{bmatrix} \, \begin{bmatrix}\mathcal{L} \left\{ x \right\} \\ \mathcal{L} \left\{ y \right\}\end{bmatrix} &= \begin{bmatrix}\frac{1}{s+2} \\ -\frac{2}{s-1}\end{bmatrix} \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 \begin{cases} \mathcal{L} \left\{ y \right\} = \left( s-1 \right) \, \mathcal{L} \left\{ x \right\} - \frac{1}{s+2} \\ -4 \mathcal{L} \left\{ x \right\} + (s+2) \, \left( (s-1) \, \mathcal{L} \left\{ x \right\} - 1 \right) = -\frac{2}{s-1} \end{cases}}
Finally we get 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\} = -\frac{2}{(s-1)(s^2 + s - 6} + \frac{1}{s^2+s-6}}
Exercise 20.3
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 X' = \begin{bmatrix}1&1&1\\2&1&-1\\-8&-5&-3\end{bmatrix} \, X}
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 \{2,-1,-2\}}
…
Exercise 11
Use method of variation of parameters to find a particular solution 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 y'' +4y'+4y = t^{-2} \, \mathrm{e}^{-2t}}
Solution to Homogeneous equation:
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 &= r^2 + 4r + 4 \\ r &\in \{-2, -2\} \\ y_1 &= \mathrm{e}^{-2t} \\ y_2 &= t \, \mathrm{e}^{-2t} \\ c_1 &= -\int \frac{g(t) \, y_2}{W\{y_1, y_2\}} \, \mathrm{d}t \\ c_2 &= \int \frac{g(t) \, y_1}{W\{y_1,y_2\}} \, \mathrm{d}t \end{align}}
Notice that now we know matrices:
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{bmatrix}y_1 & y_2 \\ y_1' & y_2' \end{bmatrix} \, \begin{bmatrix}c_1' \\ c_2' \end{bmatrix} = \begin{bmatrix}0 \\ g(t) \end{bmatrix}}