"Curve passes through the point (0,8) and has the property that the slope of the curve at every point pp is four-times the y-coordinate."
Slope of tangent line at a given point = value of derivative of function at that point.
Thus, dy/dx = derivative = 4y
Thus, dy/dx = 4y. Rearrange this with all y on one side and all x on the other:
dy
--- = 4dx
y
Integrating both sides, we get ln|y| = 4x + ln C, or ln|y| - ln C = 4x
|y|
Then ln -------- =4x. This can be written as an exponential function:
C
|y|
--- = e^(4x). Thus, |y| = C*e^(4x). Given that this curve passes thru (0,8),
C
8 = C*e^0, so C = 8. Then the function question is y = 8e^(4x).
