Solving a nonhomogeneous linear system by variation of

Chapter 8, Problem 35E

(choose chapter or problem)

Solving a nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A X}+\mathbf{F}(t)\) 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 coefficien matrix.

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

(c) Use the computer to carry out the computations of:

\(\mathbf{\Phi}^{-1}(t) \mathbf{F}(t), \quad \int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t, \quad \mathbf{\Phi}(t) \int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t\), \(\mathbf{\Phi}(t) \mathbf{C}, \text { and } \boldsymbol{\Phi}(t) \mathbf{C}+\int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t\), where C is a column matrix of constants \(c_{1}, c_{2}, c_{3}, \text { and } c_{4}\).

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

Text Transcription:

mathbf X^prime = mathbf A X+mathbf F t

mathbf X^prime = ({array} rrrr 2 & -2 & 2 & 1 -1 & 3 & 0 & 3 0 & 0 & 4 & -2 0 & 0 & 2 & -1 {array}) mathbf X+ ({array}c t e^{t} e^{-t} e^2 t 1 {array})

boldsymbol Phi t

Phi^-1 t

mathbf Phi^-1 t mathbf F t

quad int boldsymbol Phi^-1 t mathbf F t d t

quad mathbf Phi t int boldsymbol Phi^-1 t mathbf F t d t

mathbf Phi t mathbf C

boldsymbol Phi t mathbf C+int boldsymbol Phi^-1 t mathbf F t d t

c_1

c_2

c_3

c_4

mathbf X=mathbf X_c+mathbf X_p

mathbf X_c=c_1 mathbf X_1+c_2 mathbf X_2+c_3 mathbf X_3+c_4 mathbf X_4