Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.
\[ y'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2 \]
Answer:
\(F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.
\(F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.