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

question 411

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.

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In-Depth Study

A thorough and detailed examination or analysis of a particular subject or phenomenon.

Individual

A single human being as distinct from a group, class, or family, often considered in the context of individuality and personal identity.

Psychoanalytic Theory

A psychological theory originating from Sigmund Freud, emphasizing unconscious processes such as conflicts, desires, and memories.

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A length of time that is longer than usual or expected.