text: ## Exam 3 Answers
  
  10 b.
  
  The characteristic polynomial for the coefficient matrix is $\lambda^2 +2\lambda + 5$ . The roots of this are $\lambda = -1 \pm 2i$.  These are the eigenvalues of the coefficient matrix.
  
  To find the eigenvectors, solve
  
  with $\lambda=-1+2i$.
  
  We get $(2-2i)a_1-4a_2=0$.  So, if $a_1=4$ then $a_2=2-2i$.  Hence an eigenvector is $\left[\begin{array}{c} 4 \\ 2-2i \end{array}\right]$
  
  So a complex solution is
  
  So
  $x(t)=4e^{-t}(c_1\cos(2t)+c_2\sin(2t))$
  and
  $y(t)=2e^{-t}\big(c_1(\cos(2t)+\sin(2t))+c_2(-\cos(2t)+\sin(2t))\big)$
  $y(t)$ could also be written as
  $y(t)=2e^{-t}\big((c_1-c_2)\cos(2t)+(c_1+c_2)\sin(2t)\big)$
  
  Here is a slope field with a solution curve through the point $(0,1)$ at time $t=0$ with a counter-clockwise direction:
explanation: The document provides a detailed solution to a problem involving the characteristic polynomial, eigenvalues, and eigenvectors of a coefficient matrix. It also includes the general solution to the system of differential equations and a description of a slope field with a solution curve. The document does not contain a question but rather a complete solution to a problem.
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distilabel_metadata.role: system
distilabel_metadata.content: ## Exam 3 Answers
  
  10 b.
  
  The characteristic polynomial for the coefficient matrix is $\lambda^2 +2\lambda + 5$ . The roots of this are $\lambda = -1 \pm 2i$.  These are the eigenvalues of the coefficient matrix.
  
  To find the eigenvectors, solve
  
  with $\lambda=-1+2i$.
  
  We get $(2-2i)a_1-4a_2=0$.  So, if $a_1=4$ then $a_2=2-2i$.  Hence an eigenvector is $\left[\begin{array}{c} 4 \\ 2-2i \end{array}\right]$
  
  So a complex solution is
  
  So
  $x(t)=4e^{-t}(c_1\cos(2t)+c_2\sin(2t))$
  and
  $y(t)=2e^{-t}\big(c_1(\cos(2t)+\sin(2t))+c_2(-\cos(2t)+\sin(2t))\big)$
  $y(t)$ could also be written as
  $y(t)=2e^{-t}\big((c_1-c_2)\cos(2t)+(c_1+c_2)\sin(2t)\big)$
  
  Here is a slope field with a solution curve through the point $(0,1)$ at time $t=0$ with a counter-clockwise direction:
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides a detailed solution to a problem involving the characteristic polynomial, eigenvalues, and eigenvectors of a coefficient matrix. It also includes the general solution to the system of differential equations and a description of a slope field with a solution curve. The document does not contain a question but rather a complete solution to a problem.
  
  Label: no-question