the solution of the linear system $\mathbf{x} = \mathbf{a}\mathbf{x}$ is given.

$\mathbf{a} = \begin{pmatrix} -1 & -2 \\ 6 & 7 \end{pmatrix}$, $\mathbf{x}(t) = c_1 \begin{pmatrix} 1 \\ -1 \end{pmatrix} e^t + c_2 \begin{pmatrix} -10 \\ 30 \end{pmatrix} e^{5t}$

please discuss the nature of the solution in a neighborhood of $(0, 0)$.

- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -x$, then $\mathbf{x}(t)$ approaches $(0, 0)$ along this line. otherwise $\mathbf{x}(t)$ approaches $(0, 0)$ from the direction determined by $y = -3x$.
- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -3x$, then $\mathbf{x}(t)$ approaches $(0, 0)$ along this line. otherwise $\mathbf{x}(t)$ approaches $(0, 0)$ from the direction determined by $y = -x$.
- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -x$, then $\mathbf{x}(t)$ becomes unbounded along this line. otherwise $\mathbf{x}(t)$ becomes unbounded and $y = -3x$ serves as an asymptote.
- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -3x$, then $\mathbf{x}(t)$ becomes unbounded along this line. otherwise $\mathbf{x}(t)$ becomes unbounded and $y = -x$ serves as an asymptote.
- all solutions spiral toward $(0, 0)$.

with the aid of a calculator or a cas, find and graph the solution that satisfies $\mathbf{x}(0) = (1, 1)$. (enter any column vector as a row vector. select update graph to see your response plotted on the graph. select the submit button to grade your response.)

Question

the solution of the linear system $\mathbf{x} = \mathbf{a}\mathbf{x}$ is given.

$\mathbf{a} = \begin{pmatrix} -1 & -2 \\ 6 & 7 \end{pmatrix}$, $\mathbf{x}(t) = c_1 \begin{pmatrix} 1 \\ -1 \end{pmatrix} e^t + c_2 \begin{pmatrix} -10 \\ 30 \end{pmatrix} e^{5t}$

please discuss the nature of the solution in a neighborhood of $(0, 0)$.

- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -x$, then $\mathbf{x}(t)$ approaches $(0, 0)$ along this line. otherwise $\mathbf{x}(t)$ approaches $(0, 0)$ from the direction determined by $y = -3x$.
- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -3x$, then $\mathbf{x}(t)$ approaches $(0, 0)$ along this line. otherwise $\mathbf{x}(t)$ approaches $(0, 0)$ from the direction determined by $y = -x$.
- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -x$, then $\mathbf{x}(t)$ becomes unbounded along this line. otherwise $\mathbf{x}(t)$ becomes unbounded and $y = -3x$ serves as an asymptote.
- if $\mathbf{x}(0) = \mathbf{x}_0$ lies on the line $y = -3x$, then $\mathbf{x}(t)$ becomes unbounded along this line. otherwise $\mathbf{x}(t)$ becomes unbounded and $y = -x$ serves as an asymptote.
- all solutions spiral toward $(0, 0)$.

with the aid of a calculator or a cas, find and graph the solution that satisfies $\mathbf{x}(0) = (1, 1)$. (enter any column vector as a row vector. select update graph to see your response plotted on the graph. select the submit button to grade your response.)

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Phase Portrait Analysis", "Linear Systems of ODEs" ], "new_concepts": [ "Asymptotic Behavior of ODEs", "Initial Value Problems for Systems" ], "current_concepts": [ "Linear Systems of ODEs", "Phase Portrait Analysis", "Asymptotic Behavior of ODEs", "Initial Value Problems for Systems" ] } ## Analyze eigenvalues and eigenvectors Using the *Linear Systems of ODEs* knowledge point $$ \mathbf{X}(t) = c_1 \begin{pmatrix} 1 \ -1 \end{pmatrix} e^t + c_2 \begin{pmatrix} -10 \ 30 \end{pmatrix} e^{5t} $$ $$ \lambda_1 = 1 > 0, \quad \mathbf{v}_1 = \begin{pmatrix} 1 \ -1 \end{pmatrix} \implies y = -x $$ $$ \lambda_2 = 5 > 0, \quad \mathbf{v}_2 = \begin{pmatrix} -10 \ 30 \end{pmatrix} \propto \begin{pmatrix} 1 \ -3 \end{pmatrix} \implies y = -3x $$ ## Determine asymptotic behavior as t increases Using the *Phase Portrait Analysis* knowledge point $$ \text{Since } \lambda_1 > 0 \text{ and } \lambda_2 > 0, \text{ both terms grow exponentially as } t \to \infty. $$ $$ \text{If } c_2 = 0 \text{ (i.e., } \mathbf{X}(0) \text{ lies on } y = -x\text{), } \mathbf{X}(t) \text{ becomes unbounded along } y = -x. $$ $$ \text{If } c_2 e 0, \text{ since } \lambda_2 = 5 > \lambda_1 = 1, \text{ the } e^{5t} \text{ term dominates as } t \to \infty. $$ $$ \mathbf{X}(t) \approx c_2 \begin{pmatrix} -10 \ 30 \end{pmatrix} e^{5t} \implies y = -3x \text{ serves as an asymptote.} $$ ## Solve the initial value problem Using the *Linear Systems of ODEs* knowledge point $$ \mathbf{X}(0) = c_1 \begin{pmatrix} 1 \ -1 \end{pmatrix} + c_2 \begin{pmatrix} -10 \ 30 \end{pmatrix} = \begin{pmatrix} 1 \ 1 \end{pmatrix} $$ $$ \begin{cases} c_1 - 10c_2 = 1 \ -c_1 + 30c_2 = 1 \end{cases} $$ $$ 20c_2 = 2 \implies c_2 = \frac{1}{10} $$ $$ c_1 = 1 + 10c_2 = 2 $$ $$ \mathbf{X}(t) = 2 \begin{pmatrix} 1 \ -1 \end{pmatrix} e^t + \frac{1}{10} \begin{pmatrix} -10 \ 30 \end{pmatrix} e^{5t} = \begin{pmatrix} 2e^t - e^{5t} \ -2e^t + 3e^{5t} \end{pmatrix} $$ ## Question 1 If $\mathbf{X}(0) = \mathbf{X}_0$ lies on the line $y = -x$, then $\mathbf{X}(t)$ approaches $(0, 0)$ along this line. Otherwise $\mathbf{X}(t)$ approaches $(0, 0)$ from the direction determined by $y = -3x$.

If $\mathbf{X}(0) = \mathbf{X}_0$ lies on the line $y = -3x$, then $\mathbf{X}(t)$ approaches $(0, 0)$ along this line. Otherwise $\mathbf{X}(t)$ approaches $(0, 0)$ from the direction determined by $y = -x$.

If $\mathbf{X}(0) = \mathbf{X}_0$ lies on the line $y = -x$, then $\mathbf{X}(t)$ becomes unbounded along this line. Otherwise $\mathbf{X}(t)$ becomes unbounded and $y = -3x$ serves as an asymptote.

If $\mathbf{X}(0) = \mathbf{X}_0$ lies on the line $y = -3x$, then $\mathbf{X}(t)$ becomes unbounded along this line. Otherwise $\mathbf{X}(t)$ becomes unbounded and $y = -x$ serves as an asymptote.

All solutions spiral toward $(0, 0)$. ## Question 2 $$\mathbf{X}(t) = \begin{pmatrix} 2e^t - e^{5t} \ -2e^t + 3e^{5t} \end{pmatrix}$$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Linear Systems of ODEs" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze eigenvalues and eigenvectors

Determine asymptotic behavior as t increases

Solve the initial value problem

Answer:

Question 1

Question 2