Get the solution to "(a) Find a fundamental matrix for the given..."

siroticuvj

Answered question

2021-11-23

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

Answer & Explanation

Lupe Kirkland

Beginner2021-11-24Added 21 answers

The eigenvalues and related eigenvectors must first be determined. Next, determine a broad answer to the d.e. Create a basic matrix next. After that, solve the constants $c_{1}, c_{2}, c_{3}, c_{4}$ for the two conditions following from
$Φ (0) = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$x^{'} = [\begin{matrix} - 1 & - 4 \\ 1 & - 1 \end{matrix}]$
${(- 1 - λ)}^{2} + 4 = 0 \to λ^{2} + 2 λ + 5 = 0 \to λ = - 1 \pm 2 i$
$v_{1} = [\begin{matrix} 2 i \\ 1 \end{matrix}]$
$v_{2} = [\begin{matrix} - 2 i \\ 1 \end{matrix}]$
$x = [\begin{matrix} x_{1} & x_{2} \end{matrix}] c = e^{- t} [\begin{matrix} - 2 \sin (2 t) & 2 \cos (2 t) \\ \cos (2 t) & \sin (2 t) \end{matrix}] c$
$Φ (0) = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$Φ (0) = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \to$
$c_{1} = 0$
$2 c_{2} = 1$
$c_{3} = 1$
$c_{4} = 0$
$Φ (t) = e^{- t} [\begin{matrix} \cos (2 t) & - 2 \sin (2 t) \\ \frac{1}{2} \sin (2 t) & \cos (2 t) \end{matrix}]$

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