Problem 1.
(1 point)
a. Verify that $Y(t) = \begin{bmatrix} e^{4t} \\ 0 \\ e^{4t} \end{bmatrix}$ is a solution to $\vec{y}\'(t) = \begin{bmatrix} 1 & -3 & 3 \\ -3 & 1 & 3 \\ 3 & 3 & 1 \end{bmatrix} \vec{y}(t)$ by substituting it into the differential equation. (Enter the terms in the order given.)
Use angle brackets, < and >, to indicate vectors, and separate the components with commas. E.g. $(e^t, t^2, 2e^{2t}).$
b. Verify that $\vec{y}_0(t) = \begin{bmatrix} 5t+1 \\ 2t \\ 4t+2 \end{bmatrix}$ is a particular solution to
$\vec{y}\'(t) = \begin{bmatrix} 1 & -3 & 3 \\ -3 & 1 & 3 \\ 3 & 3 & 1 \end{bmatrix} \vec{y}(t) + \begin{bmatrix} 4 - (6t + 6 + 5t) \\ t - 1 \\ 2 - (21t + 3 + 4t) \end{bmatrix}$ by substituting it into the differential equation. (Enter the terms in the order given.)
Use angle brackets, < and >, to indicate vectors, and separate the components with commas. E.g. $(3, 3t^2, 2t + 1).$
