MATH 308 Lecture 36
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Section 7.9
Solving nonhomogeneous systems:
- Undetermined coefficients
- Variation of parameters
- Laplace transforms
where 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)} is a particular solution and is the general solution to the homogeneous form problem.
Exercise 1a: Undetermined Coefficients
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}1&1\\4&-2\end{bmatrix} \, X + \begin{bmatrix}\mathrm{e}^{-2t}\\-2 \mathrm{e}^{t}\end{bmatrix}}
Solution to homogeneous system 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 X_h(t) = c_1 \, \mathrm{e}^{-3t} \, \begin{bmatrix}1\\-4\end{bmatrix} + c_2 \, \mathrm{e}^{2t} \, \begin{bmatrix}1\\1\end{bmatrix}}
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 \begin{align} X_p(t) &= \mathrm{e}^{-2t} \, \vec{v} + \mathrm{e}^{t} \vec{u} \\ &= \begin{bmatrix}v_1 \, \mathrm{e}^{-2t} + u_1 \mathrm{e}^{t} \\ v_2 \mathrm{e}^{-2t} + u_2 \mathrm{e}^{t}\end{bmatrix} \end{align}}
Solve for derivative:
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_p'(t) &= -2 \mathrm{e}^{-2t} \, \vec{v} + \mathrm{e}^{t} \, \vec{u} \\ A \, X_p + G(t) &= \mathrm{e}^{-2t} \, \begin{bmatrix}1&1\\4&-2\end{bmatrix}\,\vec{v} + \mathrm{e}^{t}\,\begin{bmatrix}1&1\\4&-2\end{bmatrix}\,\vec{u} + \mathrm{e}^{t} \, \begin{bmatrix}0\\-2\end{bmatrix} + \mathrm{e}^{-2t} \, \begin{bmatrix}1\\0\end{bmatrix} \\ -2 \mathrm{e}^{-2t} \, \vec{v} + \mathrm{e}^{t} \, \vec{u} &= \mathrm{e}^{-2t} \, \begin{bmatrix}v_1 + v_2\\4v_1-2v_2\end{bmatrix} + \mathrm{e}^{t} \, \begin{bmatrix}u_1+u_2\\4u_1-2u_2\end{bmatrix}+\mathrm{e}^{t} \,\begin{bmatrix}0\\-2\end{bmatrix} + \mathrm{e}^{-2t} \, \begin{bmatrix}1\\0\end{bmatrix} \\ \end{align}}
Solve system term-wise:
Coefficients 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}v_1 + v_2 + 1 = -2v_1 \\ 4v_1 - 2v_2 = -2v_2\end{cases}}
From this we can determine that 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{v} = \left\langle 0, -1 \right\rangle}
Coefficients 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}u_1 + u_2 = u_1 \\ 4u_1 - 2u_2 - 2 = u_2\end{cases}}
From this we can determine that 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{u} = \left\langle \frac{1}{2}, 0 \right\rangle}
Exercise 1b: Variation of Parameters
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 + \begin{bmatrix}\mathrm{e}^{t} \\ t\end{bmatrix}}
Solution to homogeneous system 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 X_h(t) = c_1 \, \mathrm{e}^{t} \, \begin{bmatrix}1\\1\end{bmatrix} + c_2 \, \mathrm{e}^{-t} \, \begin{bmatrix}1\\3\end{bmatrix}}
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 \begin{align} X_p(t) &= t \, \mathrm{e}^{t} \, \vec{u} + t \, \vec{v} + \vec{w} + \mathrm{e}^{t} \, \vec{x} \\ &= \begin{bmatrix}u_1 \, t \, \mathrm{e}^{t} + t \, v_1 + w_1 + x_1 \, \mathrm{e}^{t} \\ u_2 \, t \, \mathrm{e}^{-t} + t \, v_2 + w_2 + x_2 \, \mathrm{e}^{t} \end{bmatrix} \end{align}}
Find derivative and set equal to substituted RHS
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_p'(t) &= \left( \mathrm{e}^{t} + t \, \mathrm{e}^{t} \right) \, \vec{u} + \vec{v} \\ A\,X_p(t) + G(t) &= t \, \mathrm{e}^{t} \, \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \, \vec{u} + t \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \, \vec{v} + \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \, \vec{w} + \mathrm{e}^{t} \, \begin{bmatrix}1\\0\end{bmatrix} + t \, \begin{bmatrix}0\\1\end{bmatrix} \\ \mathrm{e}^{t} \, \vec{u} + \vec{v} &= \mathrm{e}^{t} \, \begin{bmatrix}2u_1-u_2\\3u_1-2u_2\end{bmatrix} + t\begin{bmatrix}2v_1-v_2\\3v_1-2v_2\end{bmatrix}+\begin{bmatrix}2w_1-w_2\\3w_1-2w_2\end{bmatrix} + \mathrm{e}^{t} \, \begin{bmatrix}1\\0\end{bmatrix} + t \, \begin{bmatrix}0\\1\end{bmatrix} \\ & \vdots \\ X_p(t) &= t \, \mathrm{e}^{t} \, \begin{bmatrix} \frac{3}{2} \\ \frac{3}{2} \end{bmatrix} + t \, \begin{bmatrix} 1 \\ 2 \end{bmatrix} + \begin{bmatrix} 0 \\ -1 \end{bmatrix} + \mathrm{e}^{t} \, \begin{bmatrix} \frac{1}{2} \\ 0 \end{bmatrix} \end{align}}
Therefore, the general solution is the sum 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) = X_p(t) + X_h(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(t) = t \, \mathrm{e}^{t} \, \begin{bmatrix} \frac{3}{2} \\ \frac{3}{2} \end{bmatrix} + t \, \begin{bmatrix} 1 \\ 2 \end{bmatrix} + \begin{bmatrix} 0 \\ -1 \end{bmatrix} + \mathrm{e}^{t} \, \begin{bmatrix} \frac{1}{2} \\ 0 \end{bmatrix} + \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \, X + \begin{bmatrix}\mathrm{e}^{t} \\ t\end{bmatrix}}