91Ó°ÊÓ

In Exercises 1 through \(5,\) replace the given equation by a system of first- order equations. \(y^{(4)}-y=0\).

Solve the system \(X^{\prime}=A X\). $$A=\left(\begin{array}{llr}1 & 2 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 2\end{array}\right)$$

In Exercises 1 through \(7,\) find the general solution of the system \(X^{\prime}=A X\) for the given matrix A. In each case check on the linear independence of solutions by examining the Wronskian. $$ A=\left(\begin{array}{rr} 3 & 3 \\ -1 & -1 \end{array}\right) $$

Solve the system \(X^{\prime}=A X\). $$A=\left(\begin{array}{rrr}2 & 1 & -1 \\ 0 & -1 & 2 \\ 0 & 0 & -1\end{array}\right)$$

In Exercises 6 through \(9,\) replace the given system by an equivalent system of first-order equations. \(v^{\prime}-2 v+2 w^{\prime}=2-4 e^{2 x}, 2 v^{\prime}-3 v+3 w^{\prime}-w=0\)

Find the matrix requested given the following matrices: \(A=\left(\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right) \quad B=\left(\begin{array}{rr}2 & 0 \\ 1 & -1\end{array}\right) \quad C=\left(\begin{array}{rr}1 & -1 \\ 1 & 2\end{array}\right)\) \(\quad A+2 B\). \(A B+C\).

Find the general solution of the system \(X^{\prime}=A X\) for the given \(m a-\) \(\operatorname{trix} A\). $$ A=\left(\begin{array}{rr} 8 & -5 \\ 16 & -8 \end{array}\right) $$

In Exercises 1 through \(7,\) find the general solution of the system \(X^{\prime}=A X\) for the given matrix A. In each case check on the linear independence of solutions by examining the Wronskian. $$ A=\left(\begin{array}{rr} 2 & 3 \\ 1 & -2 \end{array}\right) $$

Replace the given system by an equivalent system of first-order equations. \((3 D+2) v+(D-6) w=5 e^{x},(4 D+2) v+(D-8) w=5 e^{x}+2 x-3\)

In Exercises 1 through \(7,\) find the general solution of the system \(X^{\prime}=A X\) for the given matrix A. In each case check on the linear independence of solutions by examining the Wronskian. $$ A=\left(\begin{array}{rr} 12 & -15 \\ 4 & -4 \end{array}\right) $$