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. $X_{1}=\mathrm {e} ^{8t}{\begin{bmatrix}2\\1\\2\end{bmatrix}}$
2. $X_{2}=\mathrm {e} ^{-t}{\begin{bmatrix}1\\-2\\0\end{bmatrix}}$
3. $X_{3}=\mathrm {e} ^{-t}{\begin{bmatrix}0\\-2\\1\end{bmatrix}}$

Check Linear Independence (redundant vs. complete solutions):

$W\{X_{1},X_{2},X_{3}\}={\begin{vmatrix}2\mathrm {e} ^{8t}&\mathrm {e} ^{-t}&0\\\mathrm {e} ^{8t}&-2\mathrm {e} ^{-t}&-2\mathrm {e} ^{-t}\\2\mathrm {e} ^{8t}&0&\mathrm {e} ^{-t}\end{vmatrix}}=-9\mathrm {e} ^{8t}\not \equiv 0$

Therefore, our solutions $X_{1}$, $X_{2}$, and $X_{3}$ are linearly independent; so the general solution is

$X(t)=c_{1}\,\mathrm {e} ^{8t}\,{\begin{bmatrix}2\\1\\2\end{bmatrix}}+c_{2}\mathrm {e} ^{-t}\,{\begin{bmatrix}1\\-2\\0\end{bmatrix}}+c_{3}\mathrm {e} ^{-t}\,{\begin{bmatrix}0\\-2\\1\end{bmatrix}}$

Section 7.6

Lecture Notes

Given the systems:

${\begin{aligned}X'&={\begin{bmatrix}3&-2\\4&-1\end{bmatrix}}\,X\\X'&={\begin{bmatrix}-1&-4\\1&-1\end{bmatrix}}\,X\\\end{aligned}}$

Previously, we would find the eigenvalues and eigenvectors and the general solution is of the form:

$X(t)=\sum _{i=0}^{\left|\lambda \right|}c_{i}\,\mathrm {e} ^{\lambda _{i}}\,{\vec {v}}_{i}$

where $\lambda$ represents each eigenvalue and ${\vec {v}}$ 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 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 + 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 $X=\mathrm {e} ^{t}\,\mathrm {e} ^{2i\,t}\,{\begin{bmatrix}1\\1-i\end{bmatrix}}$, 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

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) = \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: 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}

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}}