# Exponential differential equation

## Homework Statement

A curve passes through the point (0,6) 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?

## Homework Equations

Y=Yoe^kx is a solution of (dy/dx)=ky, where k is constant

## The Attempt at a Solution

I integrated to get the equation for Y above but I am not given any equation to relate x and Y. My initial thought was to set dy/dx=2ky since the slope is always twice the y value. I need to find k but I can't do that without knowing a way to relate x and Y.

## Answers and Replies

Dick
Science Advisor
Homework Helper
Doesn't "slope is always twice the y value" mean dy/dx=2y? Compare that to your relevant equation to figure out what k is.

I've tried that as well but I am still left with no way to relate Y to x.

Dick
Science Advisor
Homework Helper
Why don't you show what you tried? You get Y0 by making sure (0,6) is on the curve.

One thought I had was since I knew the slope was twice the y coordinate of point P, I said that (y-y1)/(x-x1)=2y, using 6 as y1 and 0 as x1. This gives me y=6/(1-2x). When I use this to find matching x and y, I plugged them into the Y=Yoe^kx equation and came out with a negative value for k. This can't be correct since the slope is positive. Right?

Dick
Science Advisor
Homework Helper
The slope they are talking about is the slope of the tangent line. That's just y'. You are making this too hard. Just write down the solution of y'=2y. What is it? Then try to figure out Y0. Then you are done.

Mark44
Mentor
One thought I had was since I knew the slope was twice the y coordinate of point P, I said that (y-y1)/(x-x1)=2y, using 6 as y1 and 0 as x1. This gives me y=6/(1-2x). When I use this to find matching x and y, I plugged them into the Y=Yoe^kx equation and came out with a negative value for k. This can't be correct since the slope is positive. Right?

Also, the work you did here can be used to give the equation of a straight line between (0, 6) and some point (x, y). It doesn't give you y as a function of x for the curve itself.