Question

(1 point) Consider the system of differential equations \(\vec{y}\' = \begin{bmatrix} 4 & 1\\ 1 & 4 \end{bmatrix} \vec{y}\), \(\vec{y}(0) = \begin{bmatrix} -9\\ -3 \end{bmatrix}\). Verify that \(\vec{y}(t) = c_1e^{5t}\begin{bmatrix} 1\\ 1 \end{bmatrix} + c_2e^{3t}\begin{bmatrix} 1\\ -1 \end{bmatrix}\) is a solution to the system of differential equations for any choice of the constants \(c_1\) and \(c_2\). Find values of \(c_1\) and \(c_2\) that solve the given initial value problem. (According to the uniqueness theorem, you have found the unique solution of \(\vec{y}\' = P\vec{y}\), \(\vec{y}(0) = \vec{y}_0\)). \(\vec{y}(t) = (\underline{\qquad})\cdot e^{5t}\begin{bmatrix} 1\\ 1 \end{bmatrix} + (\underline{\qquad})\cdot e^{3t}\begin{bmatrix} 1\\ -1 \end{bmatrix}\)

          (1 point) Consider the system of differential equations
\(\vec{y}\' = \begin{bmatrix} 4 & 1\\ 1 & 4 \end{bmatrix} \vec{y}\), \(\vec{y}(0) = \begin{bmatrix} -9\\ -3 \end{bmatrix}\).
Verify that
\(\vec{y}(t) = c_1e^{5t}\begin{bmatrix} 1\\ 1 \end{bmatrix} + c_2e^{3t}\begin{bmatrix} 1\\ -1 \end{bmatrix}\)
is a solution to the system of differential equations for any choice of the constants \(c_1\) and \(c_2\). Find values of \(c_1\) and \(c_2\) that solve the given initial value
problem. (According to the uniqueness theorem, you have found the unique solution of \(\vec{y}\' = P\vec{y}\), \(\vec{y}(0) = \vec{y}_0\)).
\(\vec{y}(t) = (\underline{\qquad})\cdot e^{5t}\begin{bmatrix} 1\\ 1 \end{bmatrix} + (\underline{\qquad})\cdot e^{3t}\begin{bmatrix} 1\\ -1 \end{bmatrix}\)

(1 point) Consider the system of differential equations
y⃗=́
< b m a t r i x >
y⃗, y⃗(0) =
< b m a t r i x >.
Verify that
y⃗(t) = c1e^5t
< b m a t r i x >
+ c2e^3t
< b m a t r i x >
is a solution to the system of differential equations for any choice of the constants c1 and c2. Find values of c1 and c2 that solve the given initial value
problem. (According to the uniqueness theorem, you have found the unique solution of y⃗=́ Py⃗, y⃗(0) = y⃗0).
y⃗(t) = ( )· e^5t
< b m a t r i x >
+ ( )· e^3t
< b m a t r i x >

Added by Jessica J.

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