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?

ribhu · Answer

Let (dy/dx)= Ky
on rearrangement
(dy/y)=Kdx
now integrating both the sides we get
lny=kx+c -------- (1)
since the curve passes through (2,e^2) with intercepts, 
substituting
y=e^2, x=2 and intercept 'c'= 4
we get the equation of the curve as
x+lny=2 (Answer)