Question

Curve passes through the point (0,8)(0,8) and has the property that the slope of the curve at every point pp is four-times the y-coordinate. what is the equation of the curve

Answer 1

"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).