MATH 308 Lecture 27

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Lecture Notes


April Fool's!

Section 7.1

Exercise 4

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'' + 3y' - y = \tan{x} \quad y(0) = 1 \quad y'(0) = -3}

Let 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 = y} 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 = y'} , then

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' = y' = x_2 \\ x_2' = y-3y'+\tan{x} = x_1 - 3x_2 + \tan{x}\end{cases}}

In matrix 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 \begin{bmatrix}x_1\\x_2\end{bmatrix}' = \begin{bmatrix}0&1\\1&-3\end{bmatrix} \, \begin{bmatrix}x_1 \\ x_2\end{bmatrix} + \begin{bmatrix}0 \\ \tan{x}\end{bmatrix}}

Initial Conditions

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(0) = y(0) &= 1 \\ x_2(0) = y'(0) &= -3 \end{align}}

In matrix 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 \begin{bmatrix}x_1(0)\\x_2(0)\end{bmatrix} = \begin{bmatrix}1\\-3\end{bmatrix}}

Matrix Basics

(See MATH 323 Lecture 3→)


Given 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}1&-1&-1\\2&1&0\\3&-2&1\end{bmatrix}} 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 B = \begin{bmatrix}1&1&-1\\2&-1&1\\1&1&2\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 A+B = \begin{bmatrix}2&0&-2\\4&0&1\\4&-1&3\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 3A-2B = \begin{bmatrix}1&-5&-1\\2&5&-2\\7&-8&-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 A\,B = \begin{bmatrix}-2&1&-4\\4&1&-1\\0&6&-3\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 B\,A = \begin{bmatrix}0&2&-2\\3&-5&-1\\9&-4&1\end{bmatrix}}

Just for kicks, let 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} = \begin{bmatrix}x_1\\x_2\\x_3\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 A\,\vec{x} = \begin{bmatrix}x_1-x_2-x_3\\2x_1+x_2\\3x_1-2x_2+x_3\end{bmatrix}}


Exercise 6

Verify that the given vector satisfies the 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) = \begin{bmatrix}2&-1\\3&-2\end{bmatrix} \, x(t) + \begin{bmatrix}1\\-1\end{bmatrix} \, \mathrm{e}^t \quad \quad x(t) = \begin{bmatrix}1\\0\end{bmatrix} \, \mathrm{e}^{t} + 2 \, \begin{bmatrix}1\\1\end{bmatrix} \, t \, \mathrm{e}^{t}}

Calculate the derivative of 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)} :

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) &= \begin{bmatrix}1\\0\end{bmatrix} \, \frac{\mathrm{d}}{\mathrm{d}t}\left( \mathrm{e}^{t} \right) + 2 \, \begin{bmatrix}1\\1\end{bmatrix} \, \frac{\mathrm{d}}{\mathrm{d}t}\left( t \, \mathrm{e}^{t} \right) \\ &= \frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}\mathrm{e}^{t}+2t\,\mathrm{e}^{t} \\ 0+2t\,\mathrm{e}^{t}\end{bmatrix} \\ &= \begin{bmatrix}3 \mathrm{e}^{t} + 2t\,\mathrm{e}^{t} \\ 2 \mathrm{e}^{t} + 2t\,\mathrm{e}^{t}\end{bmatrix} \end{align}}

And we find that 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)} .

Section 7.3

Lecture Notes

Given 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 \begin{align} x_1+2x_2-x_3 &= 2 \\ 2x_1+x_2+x_3 &= 1 \\ x_1-x_2+2x_3 &= -1 \end{align}}

Solve it.

(The teacher is using the long way, I'll cut to the chase...)

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{rref}\left( \begin{array}{ccc|c}1&2&-1&2\\2&1&1&1\\1&-1&2&-1\end{array} \right) = \left[ \begin{array}{ccc|c}1&0&1&0\\0&1&-1&1\\0&0&0&0\end{array} \right]}

Therefore, the system is inconsisent with 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{align} x_1 &= -x_3 \\ x_2 &= x_3+1 \\ x_3 &= x_3 \end{align}}

Or simply 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 \left\langle -\alpha, \alpha + 1, \alpha \right\rangle} 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 \alpha \in \mathbb{R}}

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