A curve with a y-intercept of 4 and passing through the point (2,e^2) has the property that the slope of the curve at any point (x,y) is directly proportional to the y-coordinate of the point. What is an equation of this curve?

Question

A curve with a y-intercept of 4 and passing through the point (2,e^2) has the property that the slope of the curve at any point (x,y) is directly proportional to the y-coordinate of the point.  What is an equation of this curve?

anonymous · Answer

The slope $y'$ is directly proportional to the value of $y$ at any given point, which means
$$y'=ky$$
for some real $k$.

This looks like the setup for solving a differential equation, so I'm going to operate under that assumption. You have two initial conditions (you'll need both).
$$y'=ky~~\iff~~\frac{dy}{y}=k\,dx~~\iff~~\int\frac{dy}{y}=k\int dx$$
Integrating and solving for $y$ yields an exponential curve. You can find the value of the constant of integration by using the first condition, and the value of $k$ by using the second.

anonymous · Answer

so $$y=e ^{kx+C}$$
separate that to $$y=e ^{kx}+e ^{C}$$
let $$e ^{C}=A$$
$$y=Ae ^{kx}$$
plug in the point (0,4) to find 
$$A=4$$
then plug in the point (2,e^2) to find
$$k=1-\ln 2$$
and the final equation should be
$$y=4e ^{(1-\ln 2)x}$$

Is that right?

anonymous · Answer

Your separation had a slight mistake, but you corrected it: $y=e^{kx+C}=e^Ce^{kx}=Ae^{kx}$.

Aside from that, yes, that's right!

anonymous · Answer

Thank you!!