Question

A curve passes through the point (0, 3) and has the property that the slope of the curve at every point p is twice the y-coordinate of p. what is the equation of the curve? (use x as the independent variable.)

Answer 1

You are given the separable differential equation
  dy/dx = 2y
and the requirement that y=3 for x=0.

The solution is found as
  ∫(dy/y) = ∫(2dx)
  ln(y) = 2x+c
Substituting the given point gives
  ln(3) = 2·0 +c
So the equation can be written as
  ln(y) = 2x+ln(3)
or
  y = 3e^(2x)