MATH 308 Lecture 30

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End Exam 3 content

Lecture Notes

Review Day for exam on Wednesday

Exercise 11

Consider the 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) = \begin{bmatrix}1&1\\4&1\end{bmatrix} \, X(t)}

1. Show that the vectors $X_{1}(t)={\begin{bmatrix}1\\2\end{bmatrix}}\,\mathrm {e} ^{3t}$ and $X_{2}(t)={\begin{bmatrix}1\\-2\end{bmatrix}}\,\mathrm {e} ^{-t}$ are solutions
2. Are they linearly independent? Describe all solutions to the system
3. Find the solution to the initial value problem $X(0)={\begin{bmatrix}3\\2\end{bmatrix}}$

Solution for 1:

${\begin{aligned}X_{1}'(t)&={\begin{bmatrix}3\mathrm {e} ^{3t}\\6\mathrm {e} ^{3t}\end{bmatrix}}\,{\begin{bmatrix}1&1\\4&1\end{bmatrix}}\,{\begin{bmatrix}\mathrm {e} ^{3t}\\2\mathrm {e} ^{3t}\end{bmatrix}}&={\begin{bmatrix}3\mathrm {e} ^{3t}\\6\mathrm {e} ^{3t}\end{bmatrix}}X_{2}'(t)&={\begin{bmatrix}-\mathrm {e} ^{-t}\\2\mathrm {e} ^{-t}\end{bmatrix}}{\begin{bmatrix}1&1\\4&1\end{bmatrix}}\,{\begin{bmatrix}\mathrm {e} ^{-t}\\-2\mathrm {e} ^{-t}\end{bmatrix}}&={\begin{bmatrix}\mathrm {e} ^{-t}\\2\mathrm {e} ^{-t}\end{bmatrix}}\end{aligned}}$

$X_{1}$ 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 solutions

Solution for 2:

Note that all solutions are 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 c_1 \, X_1 + c_2 \, X_2 = 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 W(X_1, X_2) = \begin{vmatrix}\mathrm{e}^{3t} & \mathrm{e}^{-t} \\ 2 \mathrm{e}^{3t} & -2 \mathrm{e}^{-t} \end{vmatrix} = -4 \mathrm{e}^{2t} \ne 0}

Therefore, they are linearly independent.

Solution for 3:

Solution for initial value problem satisfies 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 = 0}

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} X(t) &= c_1 \begin{bmatrix} \mathrm{e}^{3t} \\ 2 \mathrm{e}^{3t} \end{bmatrix} + c_2 \begin{bmatrix} \mathrm{e}^{-t} \\ -2 \mathrm{e}^{-t} \end{bmatrix} \\ X(0) = \begin{bmatrix}3\\2\end{bmatrix} &= c_1 \, \begin{bmatrix} 1\\2 \end{bmatrix} + c_2 \, \begin{bmatrix}1\\-2\end{bmatrix} \\ c_1 &= 2 \\ c_2 &= 1 \end{align}}

Exercise 10.2

Find the eigenvalues and eigenvectors of the given matrix 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 = \begin{bmatrix}3 & -2 \\ 4 & -1\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{align} 0 = \begin{vmatrix} 3-\lambda & -2 \\ 4 & -1-\lambda \end{vmatrix} &= (3-\lambda)(-1-\lambda) - (4)(-2) \\ 0 &= \lambda^2 - 2\lambda + 5 \\ \lambda &= 1 \pm 2i \\ \end{align}}

Eigenvector for $1+2i$

• Solve 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\,X = (1+2i)\,X} , or
• Solve 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-(1+2i))\,X = 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 \begin{align} \begin{bmatrix}2-2i & -2 \\ 4 & -2-2i\end{bmatrix} \, \begin{bmatrix}x_1 \\ x_2\end{bmatrix} &= \begin{bmatrix}0\\0\end{bmatrix} \\ \begin{bmatrix}1 & -\frac{1}{2} - \frac{1}{2}i \\ 0 & 0\end{bmatrix} \, \begin{bmatrix}x_1 \\ x_2 \end{bmatrix} &= \begin{bmatrix}0\\0\end{bmatrix} \\ \vec{x} &= \left\langle \alpha, (1-i) \alpha \right\rangle \end{align}}