MATH 308 Lecture 34

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Lecture Notes

Section 7.8

To solve a system with 1 eigenvalue with multiplicity 2:

Find space of eigenvectors for 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} .

If there are two linearly independent eigenvectors 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}_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 \vec{v}_2} , 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 X_1 = \mathrm{e}^{\lambda \, t} \, \vec{v}_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 = \mathrm{e}^{\lambda \, t} \, \vec{v}_2}

are two linearly independent solutions

If there is no pair of linearly independent eigenvectors, find an eigenvector ${\vec {v}}_{1}$, and then find a second (generalized) eigenvector satisfying

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( A-\lambda\,I \right)\,\vec{v}_2=\vec{v}_1}

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 X_1 = \mathrm{e}^{\lambda \, t} \, \vec{v}_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 = \mathrm{e}^{\lambda \, t} \, \left( t \, \vec{v}_1 + \vec{v}_2 \right)}

Exercise 2

Find general solution to 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}4&-2\\8&-4\end{bmatrix} \, X} .

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 \lambda \in \{ 0, 0 \}}

Eigenvectors: $V={\begin{bmatrix}1\\2\end{bmatrix}}$

Therefore, the only solution we have so far 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_1 = \mathrm{e}^{0t} \, \begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}1\\2\end{bmatrix}}

Let's find another solution in 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_2 = t \, \mathrm{e}^{0t} \, \begin{bmatrix}1\\2\end{bmatrix} + \mathrm{e}^{0t} \, \vec{v}_2}

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 \vec{v}_1 = \begin{bmatrix}1\\2\end{bmatrix} = (A - 0I) \, \vec{v}_2}

We find 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}_2 = \begin{bmatrix}\frac{1}{2} \, \alpha + \frac{1}{2} \\ \alpha\end{bmatrix}}

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 \alpha = 1} , and we get the vector 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}_2 = \begin{bmatrix}1\\1\end{bmatrix}}

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 X_2 = t \, \begin{bmatrix}1\\2\end{bmatrix} + \begin{bmatrix}1\\1\end{bmatrix}}

and 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 = c_1 \, \begin{bmatrix}1\\2\end{bmatrix} + c_2 \left( t \, \begin{bmatrix}1\\2\end{bmatrix} + \begin{bmatrix}1\\1\end{bmatrix} \right)}

Note: We could have taken any value for $\alpha$. The resulting general solution will look different, but the constants 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} 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 c_2} will change to compensate.

Exercise 3

Find particular solution to initioal 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' = \begin{bmatrix}1&-4\\4&-7\end{bmatrix} \, X} , where $X(0)={\begin{bmatrix}3\\2\end{bmatrix}}$

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 \lambda \in \{ -3, -3 \}}

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 \vec{v}_1 = \left\langle 1, 1 \right\rangle}

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 \left( A+3I \right)\,\vec{v}_2 = \left\langle 1,1 \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 v_2 = \left\langle \frac{1}{4}+\alpha, \alpha \right\rangle}

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 \alpha = \frac{3}{4}} , then ${\vec {v}}_{2}=\left\langle 1,{\frac {3}{4}}\right\rangle$

Therefore, our 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 = c_1 \, \mathrm{e}^{-3t} \, \begin{bmatrix}1\\1\end{bmatrix} + c_2 \, \mathrm{e}^{-3t} \, \left( t \, \begin{bmatrix}1\\1\end{bmatrix} + \begin{bmatrix}1\\\frac{3}{4}\end{bmatrix} \right)}

Plug in 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 t = 0} and solve 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 \vec{c}} :

${\begin{bmatrix}3\\2\end{bmatrix}}=c_{1}\,{\begin{bmatrix}1\\1\end{bmatrix}}+c_{2}\,{\begin{bmatrix}1\\{\frac {3}{4}}\end{bmatrix}}$

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 \vec{c} = \left\langle -1,4 \right\rangle}

So our particular 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 = -1 \mathrm{e}^{-3t} \, \begin{bmatrix}1\\1\end{bmatrix} + 4 \mathrm{e}^{-3t} \, \left( t \, \begin{bmatrix}1\\1\end{bmatrix} + \begin{bmatrix}1\\\frac{3}{4}\end{bmatrix} \right)}