<exercise masterit-seed="3676" masterit-slug="X3" masterit-name="Existence/uniqueness theorem for first-order IVPs">
<statement>
<p>
Explain what the Existence and Uniqueness Theorem for First Order IVPs
guarantees about the existence and uniqueness of solutions for the following
IVP.
</p>
<me>
y'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em}
x( -6 )= -5 </me>
</statement>
<answer>
<p><m>F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} </m> is continuous at and nearby the
initial value so a solution exists for a nearby interval.
</p>
<p><m>F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}} </m> is not continous
(or even defined)
at the initial value so the guaranteed solution may not be unique.
</p>
</answer>
</exercise>
