Question

find the general solution of the given system. (assume \\(\mathbf{x} = \

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

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

\\\

$$\begin{aligned} \\frac{dx}{dt} &= 7x + 5y \\\\ \\frac{dy}{dt} &= -2x + 9y \\end{aligned}$$

\\

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

Explanation:

<pre_analysis>
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"concepts_used": [
"Linear Systems of ODEs",
"General Solution of ODE Systems"
],
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"Complex Eigenvalues"
],
"current_concepts": [
"Linear Systems of ODEs",
"General Solution of ODE Systems",
"Complex Eigenvalues"
]
}
</pre_analysis>

<reasoning>

Set up the system matrix

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

$$\begin{pmatrix} 7 & 5 \\ -2 & 9 \end{pmatrix}$$

\]

Find the eigenvalues

Using the Linear Systems of ODEs knowledge point
\[

$$\begin{aligned} \det(\mathbf{A} - \lambda \mathbf{I}) &= \det\begin{pmatrix} 7-\lambda & 5 \\ -2 & 9-\lambda \end{pmatrix} = 0 \\ (7-\lambda)(9-\lambda) + 10 &= 0 \\ \lambda^2 - 16\lambda + 73 &= 0 \\ \lambda &= \frac{16 \pm \sqrt{256 - 292}}{2} = 8 \pm 3i \end{aligned}$$

\]

Find the eigenvector for \(\lambda = 8 + 3i\)

Using the Linear Systems of ODEs knowledge point
\[

$$\begin{aligned} (\mathbf{A} - (8+3i)\mathbf{I})\mathbf{v} &= \mathbf{0} \\ \begin{pmatrix} -1-3i & 5 \\ -2 & 1-3i \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} &= \begin{pmatrix} 0 \\ 0 \end{pmatrix} \\ (-1-3i)v_1 + 5v_2 &= 0 \implies \mathbf{v} = \begin{pmatrix} 5 \\ 1+3i \end{pmatrix} \end{aligned}$$

\]

Formulate the complex solution

Using the General Solution of ODE Systems knowledge point
\[

$$\begin{aligned} \mathbf{X}_c(t) &= e^{(8+3i)t} \begin{pmatrix} 5 \\ 1+3i \end{pmatrix} \\ &= e^{8t}(\cos(3t) + i\sin(3t)) \begin{pmatrix} 5 \\ 1+3i \end{pmatrix} \\ &= e^{8t} \begin{pmatrix} 5\cos(3t) + 5i\sin(3t) \\ (\cos(3t) - 3\sin(3t)) + i(3\cos(3t) + \sin(3t)) \end{pmatrix} \end{aligned}$$

\]

Extract real and imaginary parts

Using the General Solution of ODE Systems knowledge point
\[

$$\begin{aligned} \mathbf{X}_1(t) &= \text{Re}\{\mathbf{X}_c(t)\} = e^{8t} \begin{pmatrix} 5\cos(3t) \\ \cos(3t) - 3\sin(3t) \end{pmatrix} \\ \mathbf{X}_2(t) &= \text{Im}\{\mathbf{X}_c(t)\} = e^{8t} \begin{pmatrix} 5\sin(3t) \\ 3\cos(3t) + \sin(3t) \end{pmatrix} \\ \mathbf{X}(t) &= c_1 e^{8t} \begin{pmatrix} 5\cos(3t) \\ \cos(3t) - 3\sin(3t) \end{pmatrix} + c_2 e^{8t} \begin{pmatrix} 5\sin(3t) \\ 3\cos(3t) + \sin(3t) \end{pmatrix} \end{aligned}$$

\]

</reasoning>

<answer>
Find the general solution of the given system. (Assume \(\mathbf{X} =

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

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

\(\frac{dx}{dt} = 7x + 5y\)
\(\frac{dy}{dt} = -2x + 9y\)

\(\mathbf{X}(t) =\) <blank>\(c_1 e^{8t}

$$\begin{pmatrix} 5\cos(3t) \\ \cos(3t) - 3\sin(3t) \end{pmatrix}$$

+ c_2 e^{8t}

$$\begin{pmatrix} 5\sin(3t) \\ 3\cos(3t) + \sin(3t) \end{pmatrix}$$

\)</blank>
</answer>

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

Answer:

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"General Solution of ODE Systems",
"Complex Eigenvalues"
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</pre_analysis>

<reasoning>

Set up the system matrix

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

$$\begin{pmatrix} 7 & 5 \\ -2 & 9 \end{pmatrix}$$

\]

Find the eigenvalues

Using the Linear Systems of ODEs knowledge point
\[

$$\begin{aligned} \det(\mathbf{A} - \lambda \mathbf{I}) &= \det\begin{pmatrix} 7-\lambda & 5 \\ -2 & 9-\lambda \end{pmatrix} = 0 \\ (7-\lambda)(9-\lambda) + 10 &= 0 \\ \lambda^2 - 16\lambda + 73 &= 0 \\ \lambda &= \frac{16 \pm \sqrt{256 - 292}}{2} = 8 \pm 3i \end{aligned}$$

\]

Find the eigenvector for \(\lambda = 8 + 3i\)

Using the Linear Systems of ODEs knowledge point
\[

$$\begin{aligned} (\mathbf{A} - (8+3i)\mathbf{I})\mathbf{v} &= \mathbf{0} \\ \begin{pmatrix} -1-3i & 5 \\ -2 & 1-3i \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} &= \begin{pmatrix} 0 \\ 0 \end{pmatrix} \\ (-1-3i)v_1 + 5v_2 &= 0 \implies \mathbf{v} = \begin{pmatrix} 5 \\ 1+3i \end{pmatrix} \end{aligned}$$

\]

Formulate the complex solution

Using the General Solution of ODE Systems knowledge point
\[

$$\begin{aligned} \mathbf{X}_c(t) &= e^{(8+3i)t} \begin{pmatrix} 5 \\ 1+3i \end{pmatrix} \\ &= e^{8t}(\cos(3t) + i\sin(3t)) \begin{pmatrix} 5 \\ 1+3i \end{pmatrix} \\ &= e^{8t} \begin{pmatrix} 5\cos(3t) + 5i\sin(3t) \\ (\cos(3t) - 3\sin(3t)) + i(3\cos(3t) + \sin(3t)) \end{pmatrix} \end{aligned}$$

\]

Extract real and imaginary parts

Using the General Solution of ODE Systems knowledge point
\[

$$\begin{aligned} \mathbf{X}_1(t) &= \text{Re}\{\mathbf{X}_c(t)\} = e^{8t} \begin{pmatrix} 5\cos(3t) \\ \cos(3t) - 3\sin(3t) \end{pmatrix} \\ \mathbf{X}_2(t) &= \text{Im}\{\mathbf{X}_c(t)\} = e^{8t} \begin{pmatrix} 5\sin(3t) \\ 3\cos(3t) + \sin(3t) \end{pmatrix} \\ \mathbf{X}(t) &= c_1 e^{8t} \begin{pmatrix} 5\cos(3t) \\ \cos(3t) - 3\sin(3t) \end{pmatrix} + c_2 e^{8t} \begin{pmatrix} 5\sin(3t) \\ 3\cos(3t) + \sin(3t) \end{pmatrix} \end{aligned}$$

\]

</reasoning>

<answer>
Find the general solution of the given system. (Assume \(\mathbf{X} =

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

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

\(\frac{dx}{dt} = 7x + 5y\)
\(\frac{dy}{dt} = -2x + 9y\)

\(\mathbf{X}(t) =\) <blank>\(c_1 e^{8t}

$$\begin{pmatrix} 5\cos(3t) \\ \cos(3t) - 3\sin(3t) \end{pmatrix}$$

+ c_2 e^{8t}

$$\begin{pmatrix} 5\sin(3t) \\ 3\cos(3t) + \sin(3t) \end{pmatrix}$$

\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Calculus",
"Complex Eigenvalues"
]
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</post_analysis>