Answer each question appropriately.
-If differentiable functions $y = F ( x )$ and $y = G ( x )$ 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) F(x) = G(x) for every x in I because integrating f(x) results in one unique function.
B) 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.
C) 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.
D) There is not enough information given to determine if F(x) = G(x).

Question

Quizplus · Accepted Answer

Given an initial value problem with a specific initial condition, the solution to the differential equation is unique. This is because the initial condition allows for the determination of the integration constant, ensuring that any two functions $F(x)$ and $G(x)$ that satisfy both the differential equation and the initial condition must be identical on the interval $I$.

Answer Each Question Appropriately y=F(x)y = F ( x )y=F(x) And y=G(x)y = G ( x )y=G(x)

Answer Each Question Appropriately $y = F ( x )$ And $y = G ( x )$