Question

use the method of undetermined coefficients to solve the given nonhomogeneous system. (assume \\(\mathbf{x} = \

$$\begin{pmatrix} x \\\\ y \\end{pmatrix}$$

\\). enter any column vector as a row vector.)

\\\

$$\begin{aligned} \\frac{dx}{dt} &= 2x + 3y - 7 \\\\ \\frac{dy}{dt} &= -x - 2y + 6 \\end{aligned}$$

\\

\\(\mathbf{x}(t) =\\)

Explanation:

<pre_analysis>
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"needs_drawing": false,
"concepts_used": [
"Linear Systems of ODEs",
"General Solution of ODE Systems"
],
"new_concepts": [
"Method of Undetermined Coefficients",
"Nonhomogeneous ODE Systems"
],
"current_concepts": [
"Linear Systems of ODEs",
"General Solution of ODE Systems",
"Method of Undetermined Coefficients",
"Nonhomogeneous ODE Systems"
]
}
</pre_analysis>

<reasoning>

Find eigenvalues of the coefficient matrix

Using the Linear Systems of ODEs knowledge point
\[
A =

$$\begin{pmatrix} 2 & 3 \\ -1 & -2 \end{pmatrix}$$

\]
\[
\det(A - \lambda I) = \det

$$\begin{pmatrix} 2-\lambda & 3 \\ -1 & -2-\lambda \end{pmatrix}$$

= \lambda^2 - 1 = 0 \implies \lambda_1 = 1, \lambda_2 = -1
\]

Find eigenvectors for each eigenvalue

Using the General Solution of ODE Systems knowledge point
\[
\lambda_1 = 1: \quad (A - I)\mathbf{v}_1 = \mathbf{0} \implies

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

$$\begin{pmatrix} v_{11} \\ v_{12} \end{pmatrix}$$

=

$$\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

\implies \mathbf{v}_1 =

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

\]
\[
\lambda_2 = -1: \quad (A + I)\mathbf{v}_2 = \mathbf{0} \implies

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

$$\begin{pmatrix} v_{21} \\ v_{22} \end{pmatrix}$$

=

$$\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

\implies \mathbf{v}_2 =

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

\]
\[
\mathbf{X}_c(t) = c_1 e^t

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

+ c_2 e^{-t}

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

\]

Find particular solution using undetermined coefficients

Assume a constant particular solution vector:
\[
\mathbf{X}_p =

$$\begin{pmatrix} a \\ b \end{pmatrix}$$

\]
\[
\mathbf{X}_p' = A\mathbf{X}_p + \mathbf{F}(t) \implies

$$\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

=

$$\begin{pmatrix} 2 & 3 \\ -1 & -2 \end{pmatrix}$$

$$\begin{pmatrix} a \\ b \end{pmatrix}$$

+

$$\begin{pmatrix} -7 \\ 6 \end{pmatrix}$$

\]
\[

$$\begin{cases} 2a + 3b = 7 \\ -a - 2b = -6 \end{cases}$$

\implies a = -4, \ b = 5 \implies \mathbf{X}_p =

$$\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$

\]

Combine complementary and particular solutions

Combine the solutions:
\[
\mathbf{X}(t) = \mathbf{X}_c(t) + \mathbf{X}_p = c_1 e^t

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

+ c_2 e^{-t}

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

+

$$\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$

\]
</reasoning>

<answer>
Use the method of undetermined coefficients to solve the given nonhomogeneous system. (Assume \(\mathbf{X} =

$$\begin{pmatrix} x \\ y \end{pmatrix}$$

\). Enter any column vector as a row vector.)

\(\mathbf{X}(t) =\) <blank>\(c_1 e^t

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

+ c_2 e^{-t}

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

+

$$\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$

\)</blank>
</answer>

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

Answer:

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"concepts_used": [
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"Linear Systems of ODEs",
"General Solution of ODE Systems",
"Method of Undetermined Coefficients",
"Nonhomogeneous ODE Systems"
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}
</pre_analysis>

<reasoning>

Find eigenvalues of the coefficient matrix

Using the Linear Systems of ODEs knowledge point
\[
A =

$$\begin{pmatrix} 2 & 3 \\ -1 & -2 \end{pmatrix}$$

\]
\[
\det(A - \lambda I) = \det

$$\begin{pmatrix} 2-\lambda & 3 \\ -1 & -2-\lambda \end{pmatrix}$$

= \lambda^2 - 1 = 0 \implies \lambda_1 = 1, \lambda_2 = -1
\]

Find eigenvectors for each eigenvalue

Using the General Solution of ODE Systems knowledge point
\[
\lambda_1 = 1: \quad (A - I)\mathbf{v}_1 = \mathbf{0} \implies

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

$$\begin{pmatrix} v_{11} \\ v_{12} \end{pmatrix}$$

=

$$\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

\implies \mathbf{v}_1 =

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

\]
\[
\lambda_2 = -1: \quad (A + I)\mathbf{v}_2 = \mathbf{0} \implies

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

$$\begin{pmatrix} v_{21} \\ v_{22} \end{pmatrix}$$

=

$$\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

\implies \mathbf{v}_2 =

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

\]
\[
\mathbf{X}_c(t) = c_1 e^t

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

+ c_2 e^{-t}

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

\]

Find particular solution using undetermined coefficients

Assume a constant particular solution vector:
\[
\mathbf{X}_p =

$$\begin{pmatrix} a \\ b \end{pmatrix}$$

\]
\[
\mathbf{X}_p' = A\mathbf{X}_p + \mathbf{F}(t) \implies

$$\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

=

$$\begin{pmatrix} 2 & 3 \\ -1 & -2 \end{pmatrix}$$

$$\begin{pmatrix} a \\ b \end{pmatrix}$$

+

$$\begin{pmatrix} -7 \\ 6 \end{pmatrix}$$

\]
\[

$$\begin{cases} 2a + 3b = 7 \\ -a - 2b = -6 \end{cases}$$

\implies a = -4, \ b = 5 \implies \mathbf{X}_p =

$$\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$

\]

Combine complementary and particular solutions

Combine the solutions:
\[
\mathbf{X}(t) = \mathbf{X}_c(t) + \mathbf{X}_p = c_1 e^t

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

+ c_2 e^{-t}

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

+

$$\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$

\]
</reasoning>

<answer>
Use the method of undetermined coefficients to solve the given nonhomogeneous system. (Assume \(\mathbf{X} =

$$\begin{pmatrix} x \\ y \end{pmatrix}$$

\). Enter any column vector as a row vector.)

\(\mathbf{X}(t) =\) <blank>\(c_1 e^t

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

+ c_2 e^{-t}

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

+

$$\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$

\)</blank>
</answer>

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