AA4-9.ptx

<exercise checkit-seed="0009" 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'} - 4\right)}^{5} + {\left(3 \, {y} + t - 5\right)}^{2} = 0;\qquad
                      {y}(-2)=4
                      
                      
                  </m>
              </li>
              <li>
                  <m>
                      -e^{\left(5 \, t\right)} = {\left(t + 5\right)} {\left(t + 1\right)} y + {\left(t - 6\right)} {y'''} e^{t} - 5 \, {y''};\qquad
                      y(7)=4
                      ,y'(7)=3
                      ,y''(7)=4
                  </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>(6,+\infty)</m>.
              </li>
      </ol>
  </answer>
</exercise>