Answer Each Question Appropriately $y = \mathrm { F } ( \mathrm { x } )$

Multiple Choice

Answer each question appropriately.
-If differentiable functions $y = \mathrm { F } ( \mathrm { x } )$ and $\mathrm { y } = \mathrm { G } ( \mathrm { x } )$ both solve the initial value problem $\frac { d y } { d x } = f ( x ) , \quad y \left( x _ { 0 } \right) = y _ { 0 }$ , on an interval I, must $F ( x ) = G ( x )$ for every $x$ in I? Justify the answer.

A) There is not enough information given to determine if F(x) = G(x) .
B) F(x) = G(x) for every x in I because when given an initial condition, we can find the integration constant when integrating f(x) . Therefore, the particular solution to the initial value problem is unique.
C) F(x) and G(x) are not unique. There are infinitely many functions that solve the initial value problem. When solving the problem there is an integration constant C that can be any value. F(x) and G(x) could
Each have a different constant term.
D) F(x) = G(x) for every x in I because integrating f(x) results in one unique function.