AA4-8.ptx

<exercise checkit-seed="0008" 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 + 5\right)} {y''} e^{t} + {\left(t - 1\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} + {y'} = 0;\qquad
                      y(-7)=-4
                      ,y'(-7)=0
                      
                  </m>
              </li>
              <li>
                  <m>
                      0 = {\left({y'} - 4\right)}^{5} + {\left(3 \, {y} + t + 13\right)}^{4};\qquad
                      {y}(-4)=-2
                      
                      
                  </m>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                      By the Linear ODE Existence and Uniqueness Theorem, the IVP has
                      a unique solution defined on the interval <m>(-\infty,-5)</m>.
              </li>
              <li>
                      By the First Order ODE Existence and Uniqueness Theorem, the IVP has
                      a unique solution defined nearby the initial value.
              </li>
      </ol>
  </answer>
</exercise>