(a) Find a fundamental matrix for the given system of

siroticuvj 2021-11-23 Answered
(a) Find a fundamental matrix for the given system of equations.(b) Also find the fundamental matrix Φ(t)satisfying Φ(0)=I

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Expert Answer

Lupe Kirkland
Answered 2021-11-24 Author has 1491 answers

The problem. Find a fundamental matrix and solve for \(\displaystyle\Phi{\left({0}\right)}={I}\). First determine the eigenvalues and their corresponding eigenvectors. After that, determine a general solution of the d.e Then construct a fundamental matrix. After tha, solve the constants \(\displaystyle{c}_{{1}},{c}_{{2}},{c}_{{3}},{c}_{{4}}\) for the two conditions following from
\[\Phi(0)=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]
\[x'=\begin{bmatrix}-1&-4\\1&-1\end{bmatrix}\]
Determine the eigenvalues
\(\displaystyle{\left(-{1}-\lambda\right)}^{{2}}+{4}={0}\to\lambda^{{2}}+{2}\lambda+{5}={0}\to\lambda=-{1}\pm{2}{i}\)
Row-reduce \(A-\lambda I\) to echelon and the eigenvectors follow.
\[v_1=\begin{bmatrix}2i\\1\end{bmatrix}\]
\[v_2=\begin{bmatrix}-2i\\1\end{bmatrix}\]
Consider a general solution in the form of a fundamental matrix.
\[x=\begin{bmatrix}x_1&x_2\end{bmatrix}c=e^{-t}\begin{bmatrix}-2\sin(2t)&2\cos(2t)\\\cos(2t)&\sin(2t)\end{bmatrix}c\]
Solve the system of equations that follow from
\[\Phi(0)=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]
\[\Phi(0)=\begin{bmatrix}1&0\\0&1\end{bmatrix}\to\]
\(\displaystyle{c}_{{1}}={0}\)
\(\displaystyle{2}{c}_{{2}}={1}\)
\(\displaystyle{c}_{{3}}={1}\)
\(\displaystyle{c}_{{4}}={0}\)
Substitute the c-values and get the final result.
\[\Phi(t)=e^{-t}\begin{bmatrix}\cos(2t)&-2\sin(2t)\\\frac{1}{2}\sin(2t)&\cos(2t)\end{bmatrix}\]

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