Mathematics

Problem

Which of the following must be shown to verify that the function

y=f(x)

is a solution of the first-order differential equation

\frac{dy}{dx}=g(x,y)

on an interval

I

? a)

f(x)

is differentiable at every point of

I

. b)

f'(x)=g(x,f(x))

at a single point of

I

. c)

f'(x)=g(x,f(x))

for every

x

\,in\,

I

. d)

\displaystyle \int g(x,f(x))\,dx=f(x)+C

for some constant

C