Exponential differential equation

[w3390]

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.

 

[Dick]

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.

 

[w3390]

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

 

[Dick]

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

 

[w3390]

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]

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]

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.