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:

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"Nonhomogeneous ODE Systems"
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"new_concepts": [],
"current_concepts": [
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}
</pre_analysis>

<reasoning>

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

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

e^{-t} + c_2

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

e^t +

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

\]

Express as a single row vector

\[
\mathbf{x}(t) =

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

^T =

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

\]
</reasoning>

<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>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Calculus",
"Nonhomogeneous ODE Systems"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
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"concepts_used": [
"General Solution of ODE Systems",
"Nonhomogeneous ODE Systems"
],
"new_concepts": [],
"current_concepts": [
"General Solution of ODE Systems",
"Nonhomogeneous ODE Systems"
]
}
</pre_analysis>

<reasoning>

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

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

e^{-t} + c_2

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

e^t +

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

\]

Express as a single row vector

\[
\mathbf{x}(t) =

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

^T =

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

\]
</reasoning>

<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>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Calculus",
"Nonhomogeneous ODE Systems"
]
}
</post_analysis>