(a) Use (1) to find the general solution of X a 4 2 3 3b X. Use a CAS to find eAt . Then

Chapter 10, Problem 27

(choose chapter or problem)

(a) Use (1) to find the general solution of \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right) \mathbf{X}\). Use a CAS to find \(e^{\mathbf{A} t}\). Then use the computer to find eigenvalues and eigenvectors of the coefficient matrix \(\mathbf{A}=\left(\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right)\) and form the general solution in the manner of Section 10.2. Finally, reconcile the two forms of the general solution of the system.

(b) Use (1) to find the general solution of \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}-3 & -1 \\ 2 & -1\end{array}\right) \mathbf{X}\). Use a CAS to find \(e^{\mathbf{A} t}\). In the case of complex output, utilize the software to do the simplification; for example, in Mathematica, if m=MatrixExp[At] has complex entries, then try the command Simplify[ComplexExpand[m]].