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

(a) Find a fundamental matrix for the given system of equations.(b) Also find the fundamental matrix Φ(t)satisfying Φ(0)=I

• Questions are typically answered in as fast as 30 minutes

Plainmath recommends

• Get a detailed answer even on the hardest topics.
• Ask an expert for a step-by-step guidance to learn to do it yourself.

Lupe Kirkland

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.
$\Phi(t)=e^{-t}\begin{bmatrix}\cos(2t)&-2\sin(2t)\\\frac{1}{2}\sin(2t)&\cos(2t)\end{bmatrix}$