AA4-7.ptx

<exercise checkit-seed="0007" 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>
                      0 = -{\left(2 \, {y} + t + 9\right)}^{7} - {\left({y'} - 5\right)}^{5};\qquad
                      {y}(-2)=-3
                      
                      
                  </m>
              </li>
              <li>
                  <m>
                      {\left(t + 4\right)} {\left(t + 1\right)} y + 4 \, {y''} + e^{t} = -{\left(t - 6\right)} {y'''} e^{t};\qquad
                      y(3)=8
                      ,y'(3)=6
                      ,y''(3)=2
                  </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>(-\infty,6)</m>.
              </li>
      </ol>
  </answer>
</exercise>