2. Using vector notation for a system of linear homogeneous first order ODE, $d\vec{X}/dt = A \cdot \vec{X}$, obtain the general solution when the matrix A is given by (a) $A = \begin{pmatrix} -1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 3 & -1 \end{pmatrix}$ (b) $A = \begin{pmatrix} -6 & 5 \\ -5 & 4 \end{pmatrix}$ (c) $A = \begin{pmatrix} 6 & -1 \\ 5 & 2 \end{pmatrix}$
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Substituting this into the equation, we get: [d/dt x; d/dt y] = [a b; c d] [x; y] Expanding this equation, we get: [d/dt x = ax + by; d/dt y = cx + dy] Now, we can write these equations as a system of first-order ODEs: dx/dt = ax + by dy/dt = cx + dy To Show more…
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