Acurve passes through the point (0, 2) 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

You have the separable differential equation
  y' = 2y
with the boundary condition
  y(0) = 2

It can be solved by integrating
  ∫(dy/y) = ∫(2dx)
  ln(y) = 2x +c

The boundary condition tells you
  ln(2) = 2·0 +c
so the constant is
  c = ln(2)
and the equation is
  ln(y) = 2x + ln(2)

Taking antilogs, we have
  y = 2·e^(2x)