<exercise masterit-seed="8387" 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'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em}
x( 3 )= -5 </me>
</statement>
<answer>
<p><m>F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} </m> is continuous at and nearby the
initial value so a solution exists for a nearby interval.
</p>
<p><m>F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}} </m> is continous
at and nearby the initial value so the solution is unique for
a nearby interval.
</p>
</answer>
</exercise>
