91Ó°ÊÓ

In Problems 21-30, find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & -1 & 1 \end{array}\right) \mathbf{X} $$

In Problems 13-32, use vaniation of parameters to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right) \mathbf{X}+\left(\begin{array}{c} 1 \\ \cot t \end{array}\right) $$

Use (1) to find the general solution of $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrrr} -4 & 0 & 6 & 0 \\ 0 & -5 & 0 & -4 \\ -1 & 0 & 1 & 0 \\ 0 & 3 & 0 & 2 \end{array}\right) \mathbf{X} $$ Use a CAS to find \(e^{\mathrm{A} t}\).

In Problems 13-32, use vaniation of parameters to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} 1 & 2 \\ -\frac{1}{2} & 1 \end{array}\right) \mathbf{X}+\left(\begin{array}{c} \csc t \\ \sec t \end{array}\right) e^{t} $$

In Problems 21-30, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 2 & -1 \\ 0 & 1 & 0 \end{array}\right) X $$

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 2 & -1 \\ 0 & 1 & 0 \end{array}\right) \mathbf{X} $$

Use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & 2 \\ -\frac{1}{2} & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{l}\csc t \\ \sec t\end{array}\right) \boldsymbol{e}^{t}\)

In Problems 21-30, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{lll} 4 & 1 & 0 \\ 0 & 4 & 1 \\ 0 & 0 & 4 \end{array}\right) X $$

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{lll} 4 & 1 & 0 \\ 0 & 4 & 1 \\ 0 & 0 & 4 \end{array}\right) \mathbf{X} $$

Use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}1 & -2 \\ 1 & -1\end{array}\right) \mathbf{x}+\left(\begin{array}{c}\tan t \\\ 1\end{array}\right)\)