Mathematics

Problem

A family of solutions to the differential equation

\dfrac{dy}{dx}=x y^{2}

can be expressed implicitly as

y^{-1}= -\dfrac{x^{2}}{2}+C

, where

C

is a constant. Which of the following statements best describes the effect of applying an initial condition such as

y(0)=3

to this family? a) Because

x=0

in the initial condition, it forces

C=0

for every solution, so the constant disappears. b) It determines the numerical value of

C

, selecting exactly one curve from the family. c) It shifts the entire family of curves vertically without changing the value of

C

. d) It fixes the slope of each curve at

x=0

, but several different curves can still share that slope, so more than one solution remains.