MATH 308 Lecture 37

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Lecture Notes

Section 7.9

Exercise 1c: Imaginary Eigenvectors

$X^{\prime }(t)=\left[\begin{array}{cc}2& -5\\ 1& -2\end{array}\right]X+\left[\begin{array}{c}-\cos t\\ \sin t\end{array}\right]$

Eigenvalues: $\lambda \in \{-i,i\}$

$X_h(t)=c_1({\overrightarrow{v}}_1\cos t-{\overrightarrow{v}}_2\sin t)+c_2({\overrightarrow{v}}_2\cos t+{\overrightarrow{v}}_1\sin t)$

Guess for $X_p(t)$:

$X-p(t)=t\overrightarrow{a}\cos t+t\overrightarrow{b}\sin t+\overrightarrow{c}\cos t+\overrightarrow{d}\sin t$

Exercise 2a: Laplace Transform

$X^{\prime }(t)=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]X+\left[\begin{array}{c}t\\ -1\end{array}\right]X(0)=\left[\begin{array}{c}2\\ 1\end{array}\right]$

$\{\begin{array}{l}{ℒ}\left\{x^{\prime }\right\}={ℒ}\left\{y\right\}+{ℒ}\left\{t\right\}\\ {ℒ}\left\{y^{\prime }\right\}={ℒ}\left\{x\right\}+{ℒ}\{-1\}\end{array}$

$\{\begin{array}{l}s{ℒ}\left\{x\right\}-x(0)={ℒ}\left\{y\right\}+\frac{1}{s^2}\\ s{ℒ}\left\{y\right\}-y(0)={ℒ}\left\{x\right\}+\frac{1}{s}\end{array}$

$\{\begin{array}{l}s{ℒ}\left\{x\right\}-2={ℒ}\left\{y\right\}+\frac{1}{s^2}\\ s{ℒ}\left\{y\right\}-1={ℒ}\left\{x\right\}+\frac{1}{s}\end{array}$

${ℒ}\left\{x\right\}=s{ℒ}\left\{y\right\}-1+\frac{1}{s}$

${ℒ}\left\{y\right\}=\frac{1}{s^2(s^2-1)}+\frac{1}{s^2-1}+\frac{s}{s^2-1}$

${ℒ}\left\{x\right\}=s(\frac{1}{s^2(s^2-1)}+\frac{1}{s^2-1}+\frac{s}{s^2-1})-1+\frac{1}{s}$

Take inverse Laplace transforms

Exercise 2b

Using matrix notation,

$X^{\prime }(t)=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]X(t)+\left[\begin{array}{c}1\\ -1\end{array}\right]{\mathrm{e}}^{t}X(0)=\left[\begin{array}{c}3\\ 0\end{array}\right]$

$\begin{array}{rl}{ℒ}\left\{X^{\prime }\right\}& =\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]{ℒ}\left\{X\right\}+\left[\begin{array}{c}1,-1\end{array}\right]{ℒ}\left\{{\mathrm{e}}^t\right\}\\ s{ℒ}\left\{X\right\}-X(0)& =\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]{ℒ}\left\{X\right\}+\left[\begin{array}{c}1\\ -1\end{array}\right]\left(\frac{1}{s-1}\right)\\ (sI-\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]){ℒ}\left\{X\right\}& =X(0)+\frac{1}{s-1}\left[\begin{array}{c}1\\ -1\end{array}\right]\\ {ℒ}\left\{X\right\}& ={\left[\begin{array}{cc}s-2& 1\\ -3& s+2\end{array}\right]}^{-1}(X(0)+\frac{1}{s-1}\left[\begin{array}{c}1\\ -1\end{array}\right])\\ {ℒ}\left\{X\right\}& =\frac{1}{s^2-1}\left[\begin{array}{cc}s+2& -1\\ 3& s-2\end{array}\right]\left[\begin{array}{c}3+\frac{1}{s-1}\\ -\frac{1}{s-1}\end{array}\right]\end{array}$