MATH 308 Lecture 38
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Variation of Parameters in Systems
- 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_1} and 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_2} are two linearly independent solutions to homogeneous equation
- Find a particular solution to nonhomogeneous system 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) = c_1(t) \, X_1 + c_2(t) \, X_2} , 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 c_1} and 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} are real-valued functions.
Taking the derivative of the equation 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(t)} , 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 X'(t) = c_1'(t) \, X_1 + c_2'(t) \, X_2 + c_1(t) \, X_1' + c_2(t) \, X_2'}
By the differential equation, this has to be equal to
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 A \, \left( c_1 \, X_1 + c_2 \, X_2 \right) + G(t)}
Notice that we have an equation 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_1' = A \, X_1} and 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_2' = A \, X_2}
Therefore,
Exercise 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'(t) = \begin{bmatrix}4 &-2 \\ 8 & -4 \end{bmatrix} X(t) + \begin{bmatrix}t^{-3} \\ -t^{-2}\end{bmatrix}}
Use variation of parameters, but first solve homogeneous equation first:
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 \{0,0\}}
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 \alpha, 2\alpha \right\rangle}
Generalized Eigenvector: 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}_2 = \left\langle 0, - \frac{1}{2} \right\rangle}
Homogeneous solutions: 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}X_1 = \left\langle 1,2 \right\rangle \\ X_2 = t \, \left\langle 1, 2 \right\rangle - \left\langle 0, \frac{1}{2} \right\rangle\end{cases}}
Particular solution is of form 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 = c_1(t) \, X_1 + c_2(t) \, X_2} , 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 c_1' \, X_1 + c_2' \, X_2 = G(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 c_1' \begin{bmatrix}1\\2\end{bmatrix} + c_2' \begin{bmatrix}t\\2t-\frac{1}{2}\end{bmatrix} = \begin{bmatrix} t^{-3}\\-t^{-2}\end{bmatrix}}
From this, 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 \begin{cases}c_1' = t^{-3} + 4t^{-2} + 2t^{-1} \\ c_2' = 4t^{-3} - 2t^{-2}\end{cases}}
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} c_1 &= -\frac{t^{-2}}{2} + 4t^{-1} - 2 \ln{t} \\ c_2 &= -\frac{4t^{-2}}{2} - 2t^{-1} \end{align}}
Extra Material: Chapter 5
not on final
Taylor series:
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} f(t) &= \sum_{i=0}^\infty \frac{f^{(i)}(0)}{i!} \, x^i \\ &= \frac{f(0)}{0!} \, x^0 + \frac{f'(0)}{1!} \, x + \frac{f''(0)}{2!} \, x^2 + \dots + \frac{f^{(n)}(c)}{n!} \end{align}}
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 c \in (0,x)}
Suppose we had a differential 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 y'' - y = 0} 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 y = a_0 + a_1 \, x + a_2 \, x^2 + \dots + a_n \, x^n}
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} y' &= a_1 + 2a_2 \, x + 3a_3 \, x^2 + \dots + n \, a_n \, x^{n-1} y'' &= 2a_2 + 3 \cdot 2 \, a_3 \, x + 4 \cdot 3 \, a_4 \, x^2 + \dots + n(n-1)a_n \, x^{n-2} \end{align}}
in other words
Suppose we had a differential 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 y'' - y = 0} 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 y = \sum_{k=0}^\infty a_k \, x^{k}}
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} y' &= \sum_{k=0}^\infty (k+1) \, a_{k+1} \, x^k \\ y'' &= \sum_{k=0}^\infty (k+1)(k+2) \, a_{k+2} \, x^k \end{align}}
By the differential equation, for any 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 k \ge 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 (k+2)(k+1) \, a_{k+2} = -a_k}