Solved: Solving a nonhomogeneous linear system X AX F(t) by variation of parameters when

Chapter 10, Problem 36

(choose chapter or problem)

Solving a nonhomogeneous linear system \(\mathbf{X}^\prime=\mathbf{AX+F}(t)\) by variation of parameters when A is a 3 X 3 (or larger) matrix is almost an impossible task to do by hand. Consider the system

\(\mathbf{X}^{\prime}=\left(\begin{array}{rrrr} 2 & -2 & 2 & 1 \\ -1 & 3 & 0 & 3 \\ 0 & 0 & 4 & -2 \\ 0 & 0 & 2 & -1 \end{array}\right) \mathbf{X}+\left(\begin{array}{c} t e^{t} \\ e^{-t} \\ e^{2 t} \\ 1 \end{array}\right)\).

(a) Use a CAS or linear algebra software to find the eigenvalues and eigenvectors of the coefficient matrix.

(b) Form a fundamental matrix \(\boldsymbol{\Phi}(t)\) and use the computer to find \(\boldsymbol{\Phi}^{-1}(t)\).

(c) Use the computer to carry out the computations of \(\boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t), \int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t, \boldsymbol{\Phi}(t) \int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t, \boldsymbol{\Phi}(t) \mathbf{C}\), and \(\boldsymbol{\Phi}(t) \mathbf{C}+\int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t\), where C is a column matrix of constants \(c_{1}, c_{2}, c_{3}\), and \(c_{4}\).

(d) Rewrite the computer output for the general solution of the system in the form \(\mathbf{X}=\mathbf{X}_{c}+\mathbf{X}_{p}\), where \(\mathbf{X}_{c}=c_{1} \mathbf{X}_{1}+c_{2} \mathbf{X}_{2}+c_{3} \mathbf{X}_{3}+c_{4} \mathbf{X}_{4}\).