G "a curve passes through the point" 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
Unfortunately, you have not shared the point through which the curve passes. Would you please do that now.
Just supposing that the graph passes through the point (2,2) (which I have invented as an example):
Write the differential equation dy/dx = 2y. Rewrite this as dy/y=2dx. Integrating both sides, ln|y|=2x+ln|c| (where c is just a constant of integration).
Solving for y: ln|y|-ln|c|=2x, or ln|y/c|=2x
then y/c=e^(2x), or y=c*e^(2x). What is the value of c? To determine this, let x=2 and y=2:
2=c*e^(2[2]) after substituting the coordinates of the point (2,2). Then
2=ce^4, or c=1/[e^4].
Substituting this c into the solution,
y= (1/[e^4])e^[2x]
This solution can be used as is, or you could try simplifying it. \ Note that if your graph goes through some point other than (2,2), the correct answer to this problem will be different.
