AA4-5.ptx

<exercise checkit-seed="0005" 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'} + 5\right)}^{3} = -{\left(3 \, {y} + t - 3\right)}^{4};\qquad
                      {y}(-6)=4
                      
                      
                  </m>
              </li>
              <li>
                  <m>
                      -{\left(t + 5\right)} {\left(t + 1\right)} - {\left(t^{2} + 25\right)} y = {\left(t - 5\right)} {y''} + 2 \, {y'};\qquad
                      y(7)=3
                      ,y'(7)=-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>(5,+\infty)</m>.
              </li>
      </ol>
  </answer>
</exercise>