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 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(t) = \begin{bmatrix}1\\2\end{bmatrix} \, \mathrm{e}^{3t}} 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(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 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(0) = \begin{bmatrix}3\\2\end{bmatrix}}


Solution for 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} 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 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 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}}

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