Question

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

Answer 1

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.
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Note that if your graph goes through some point other than (2,2), the correct answer to this problem will be different.