AA2-6.ptx

<exercise checkit-seed="0006" checkit-slug="AA2" checkit-title="Direction Fields">
  <statement>
      <p>
Modify the line of SageMath code
<c>t,y = var('t y'); plot_slope_field(FIXME, (t,-5,5), (y,-5,5))</c>
and run it at <url href="https://sagecell.sagemath.org/"/> to
produce the direction field for the ODE <m>{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t</m>.
      </p>
      <p>
Then explain how to use its direction field to approximate the
value of <m>y(-2)</m> given
the initial value <m>y(0)=1</m>.
      </p>
  </statement>
  <answer>
      <p>
          Replace <c>FIXME</c> with <c>t*y/9-t/3</c> to show
          the direction field. Then sketch a curve passing through the initial value
          to show that
          <m>y(-2)\approx 0.58</m>.
      </p>
  </answer>
</exercise>