AA4-0.ptx

<exercise checkit-seed="0000" 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(t + 3 \, {y} - 11\right)}^{7} + {\left({y'} + 3\right)}^{5} = 0;\qquad
                      {y}(6)=2
                      
                      
                  </m>
              </li>
              <li>
                  <m>
                      -e^{\left(2 \, t\right)} = {\left(t + 5\right)} {\left(t - 5\right)} y + {\left(t - 2\right)} {y''} e^{t} - 2 \, {y'};\qquad
                      y(6)=-1
                      ,y'(6)=-6
                      
                  </m>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                      By the First Order ODE Existence and Uniqueness Theorem, the IVP has
                      a unique solution defined for all real numbers.
              </li>
              <li>
                      By the Linear ODE Existence and Uniqueness Theorem, the IVP has
                      a unique solution defined on the interval <m>(2,+\infty)</m>.
              </li>
      </ol>
  </answer>
</exercise>