MATH 308 Lecture 32

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Lecture Notes

Quiz over Ch. 7.5 Wednesday

Section 7.5

Exercise 5

Find the general solution of the given system of equations:

$X'={\begin{bmatrix}3&2&4\\2&0&2\\4&2&3\end{bmatrix}}\,X$

Eigenvalues: $\lambda \in \{8,-1,-1\}$

Eigenvectors:

1. $\lambda _{1}=8:\left\langle 2,1,2\right\rangle$
2. $\lambda _{2}=-1:\left\langle \alpha ,-2\alpha -2\beta ,\beta \right\rangle =\alpha \left\langle 1,-2,0\right\rangle +\beta \left\langle 0,-2,1\right\rangle$ (note we have two eigenvectors)

Corresponding Solutions:

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_1 = \mathrm{e}^{8t} \begin{bmatrix}2\\1\\2\end{bmatrix}}
2. 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 = \mathrm{e}^{-t} \begin{bmatrix}1\\-2\\0\end{bmatrix}}
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_3 = \mathrm{e}^{-t} \begin{bmatrix}0\\-2\\1\end{bmatrix}}

Check Linear Independence (redundant vs. complete 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 W\{X_1, X_2, X_3\} = \begin{vmatrix}2\mathrm{e}^{8t} & \mathrm{e}^{-t} & 0 \\ \mathrm{e}^{8t} & -2 \mathrm{e}^{-t} & -2 \mathrm{e}^{-t} \\ 2 \mathrm{e}^{8t} & 0 & \mathrm{e}^{-t}\end{vmatrix} = -9 \mathrm{e}^{8t} \not\equiv 0}

Therefore, our 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 X_1} , $X_{2}$, 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_3} are linearly independent; so the general solution 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(t) = c_1 \, \mathrm{e}^{8t} \, \begin{bmatrix}2\\1\\2\end{bmatrix} + c_2 \mathrm{e}^{-t} \, \begin{bmatrix}1\\-2\\0\end{bmatrix} + c_3 \mathrm{e}^{-t} \, \begin{bmatrix}0\\-2\\1\end{bmatrix}}

Section 7.6

Lecture Notes

Given the 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 \begin{align} X' &= \begin{bmatrix}3&-2\\4&-1\end{bmatrix} \, X \\ X' &= \begin{bmatrix}-1&-4\\1&-1\end{bmatrix} \, X \\ \end{align}}

Previously, we would find the eigenvalues and eigenvectors and the general solution is of the 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(t) = \sum_{i = 0}^{\left|\lambda\right|} c_i \, \mathrm{e}^{\lambda_i} \, \vec{v}_i}

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 \lambda} represents each eigenvalue 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 \vec{v}} represents the corresponding eigenvector.

What about matrices with imaginary 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 \begin{align} \begin{vmatrix}3-\lambda&-2\\4&-1-\lambda\end{vmatrix} &= 0 \\ \lambda^2 - 2\lambda + 5 &= 0 \\ \lambda &= 1 \pm 2i \end{align}}

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 \lambda = 1 + 2i : \left\langle 1, 1-i \right\rangle}
2. We don't need to consider the second eigenvalue since they are complex conjugates

Recall from Chapter 3 that a root $a+bi$ has real-valued 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 c_1 \, \mathrm{e}^{a\,t} \, \cos{b\,t} + c_2 \, \mathrm{e}^{a\,t}\,\sin{b\,t}} . Let's do the same here:

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 = \mathrm{e}^{t} \, \mathrm{e}^{2i\,t} \, \begin{bmatrix}1\\1-i\end{bmatrix}} , 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 \mathrm{e}^{t} \, \left( \cos{2t} + i \, \sin{2t} \right) \left( \begin{bmatrix}1\\1\end{bmatrix} + i \begin{bmatrix}0\\-1\end{bmatrix} \right)}

Expanding this gives

$X(t)=\underbrace {\left(\mathrm {e} ^{t}\,\cos {2t}\,{\begin{bmatrix}1\\1\end{bmatrix}}-\mathrm {e} ^{t}\,\sin {2t}\,{\begin{bmatrix}0\\-1\end{bmatrix}}\right)} _{P}+i\,\underbrace {\left(\mathrm {e} ^{t}\,\sin {2t}\,{\begin{bmatrix}1\\1\end{bmatrix}}+\mathrm {e} ^{t}\,\cos {2t}\,{\begin{bmatrix}0\\-1\end{bmatrix}}\right)} _{Q}$

The real part of this 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 P} and the imaginary part of this 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 Q} are each solutions to the original system, and they are always linearly independent

Theorem

Given a system of differential equations

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) = A\,X(t)}

If 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 r_1 = \lambda + i \, \mu} 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 r_2 = \lambda - i \, \mu} is a pair of complex conjugate eigenvalues 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 \xi = \vec{a} \pm i \, \vec{b}} is the corresponding pair of eigenvectors, then the vectors

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} \vec{u} &= \mathrm{e}^{\lambda \, t} \, \left( \vec{a} \, \cos{\mu \, t} - \vec{b} \, \sin{\mu \, t} \right) \\ \vec{v} &= \mathrm{e}^{\lambda \, t} \, \left( \vec{a} \, \sin{\mu \, t} + \vec{b} \, \cos{\mu \, t} \right) \end{align}}

are real valued solutions of the system

Example

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&-4\\1&-1\end{bmatrix} \, X}

Eigenvalue: $\lambda =-1+2i$

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 \begin{bmatrix}2i\\1\end{bmatrix} = \begin{bmatrix}0\\1\end{bmatrix} + i \begin{bmatrix}2\\0\end{bmatrix}}

Apply the theorem:

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_1 &= \mathrm{e}^{-t} \, \cos{2t} \, \begin{bmatrix} 0 \\ 1 \end{bmatrix} - \mathrm{e}^{-t} \, \sin{2t} \, \begin{bmatrix} 2\\0\end{bmatrix} \\ X_2 &= \mathrm{e}^{-t} \, \cos{2t} \, \begin{bmatrix}2\\0\end{bmatrix} + \mathrm{e}^{-t} \, \sin{2t} \, \begin{bmatrix}0\\1\end{bmatrix} \end{align}}