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

question 285

Multiple Choice

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).

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