AA4-4.ptx

<exercise checkit-seed="0004" checkit-slug="AA4" checkit-title="Existence/uniqueness IVP Theorems">
  <statement>
      <p>
          Explain how to use an appropriate Existence and Uniqueness Theorem to determine
          the largest possible domain guaranteed for a unique solution to each IVP.
      </p>
      <ol>
              <li>
                  <m>
                      -{\left({y'} - 2\right)}^{5} - {\left(2 \, {y} + t - 14\right)}^{2} = 0;\qquad
                      {y}(5)=6
                      
                      
                  </m>
              </li>
              <li>
                  <m>
                      {\left(t + 4\right)} {\left(t + 1\right)} y + e^{\left(-2 \, t\right)} = -{\left(t - 5\right)} {y'''} e^{t} + 5 \, {y''};\qquad
                      y(9)=1
                      ,y'(9)=-4
                      ,y''(9)=2
                  </m>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                      By the First Order ODE Existence and Uniqueness Theorem, the IVP has
                      a unique solution defined nearby the initial value.
              </li>
              <li>
                      By the Linear ODE Existence and Uniqueness Theorem, the IVP has
                      a unique solution defined on the interval <m>(5,+\infty)</m>.
              </li>
      </ol>
  </answer>
</exercise>