Question 3.7.1: Find the general solution of the nonhomogeneous system given...

From our determination of the eigenvalues and eigenvectors of the same coefficient matrix in example 3.6.2, the complementary solution is

x_{h} = c_{1}e^{−t}\begin{bmatrix}1 \\ 3\end{bmatrix} c_{2}e^{t}\begin{bmatrix}1 \\ 1\end{bmatrix}

Therefore, the fundamental matrix is

Φ(t ) =\begin{bmatrix}e^{−t} &e^{t} \\ 3e^{−t}&e^{t}\end{bmatrix}

According to (3.7.10), we next need to compute Φ^{−1}. While the inverse of this matrix of functions may be computed by row-reducing [ Φ| I] in the usual way, because of the function coefficients in Φ it is much easier to use a shortcut for computing the inverse of a 2× 2 matrix that we established in exercise 19 of section 1.9. Specifically, if

A =\begin{bmatrix} a&b\\ c& d \end{bmatrix}

is an invertible matrix, then

A^{−1} = \frac {1}{det(A)} \begin{bmatrix} d&-b\\ -c&a \end{bmatrix}

Here, since det(Φ) = e^{−t} e^{t} −3e^{−t} e^{t} =−2, it follows

Φ^{−1}=-\frac {1}{2} \begin{bmatrix}e^{t} &-e^{t} \\ -3e^{−t}&e^{-t}\end{bmatrix}

Thus, by (3.7.10), we now have

x_{p}(t ) = Φ(t) \int {Φ^{−1}(t)b(t ) dt}         (3.7.10)

=\begin{bmatrix}e^{−t} &e^{t} \\ 3e^{−t}&e^{t}\end{bmatrix} \int { \begin{bmatrix}-\frac {1}{2}e^{t} &-\frac {1}{2}-e^{t} \\ \frac {3}{2}e^{−t}&-\frac {1}{2}e^{-t}\end{bmatrix} \begin{bmatrix} 0\\ 4t \end{bmatrix}dt}

=\begin{bmatrix}e^{−t} &e^{t} \\ 3e^{−t}&e^{t}\end{bmatrix}\int {\begin{bmatrix} 2te^{t}\\ 2te^{−t}\end{bmatrix}dt}

Integrating the vector function component-wise by parts and computing the subsequent matrix product,

x_{p}(t )=\begin{bmatrix}e^{−t} &e^{t} \\ 3e^{−t}&e^{t}\end{bmatrix} \begin{bmatrix} 2(t-1)e^{t}\\ 2(t+1)e^{−t}\end{bmatrix}

=\begin{bmatrix} 2(t −1)+2(t +1)\\ 6(t −1)+2(t +1)\end{bmatrix}

=\begin{bmatrix} 4t\\ 8t −4\end{bmatrix}

Therefore, the general solution to the original nonhomogeneous system is

x = x_{h} +x_{p} = c_{1}e^{−t}\begin{bmatrix}1 \\ 3\end{bmatrix} c_{2}e^{t}\begin{bmatrix}1 \\ 1\end{bmatrix} + \begin{bmatrix} 4t\\ 8t −4\end{bmatrix}