Question

found the following complementary function and particular solution for the given differential equation.

\\(\mathbf{x}_c = c_1 \

$$\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}$$

e^{-t} + c_2 \

$$\begin{pmatrix} -3 \\\\ 1 \\end{pmatrix}$$

e^t\\)

\\(\mathbf{x}_p = \

$$\begin{pmatrix} -2 \\\\ 3 \\end{pmatrix}$$

\\)

use the fact that \\(\mathbf{x} = \mathbf{x}_c + \mathbf{x}_p\\) is the general solution of the nonhomogeneous system of differential equations to solve. (enter any column vector as a row vector.)

Explanation:

Combine complementary and particular solutions

$$ \mathbf{x}(t) = \mathbf{x}_c(t) + \mathbf{x}_p(t) $$

Substitute given vector functions

$$ \mathbf{x}(t) = c_1
LATEXBLOCK0
e^{-t} + c_2
LATEXBLOCK1
e^t +
LATEXBLOCK2
$$

Express as a single row vector

$$ \mathbf{x}(t) =
LATEXBLOCK3
^T =
LATEXBLOCK4
$$

Answer:

... use the fact that \(\mathbf{x} = \mathbf{x}_c + \mathbf{x}_p\) is the general solution of the nonhomogeneous system of differential equations to solve. (Enter any column vector as a row vector.)

<blank>\(

$$\begin{pmatrix} c_1 e^{-t} - 3 c_2 e^t - 2, & -c_1 e^{-t} + c_2 e^t + 3 \end{pmatrix}$$

\)</blank>