A curve passes through the point (7,6) and has the property that the slope of the curve at every point P is 4 times the y-coordinate of P. What is the equation of the curve? Simplify the equation as much as possible.

Question

MathMate · Answer

dy/dx = 4y
separate variables,
dy/y = 4dx
Integrate
log(y)=4x+C
or y=Ce^4x
At x=7, y=6, so
6=Ce^(4*7)
C=6/e^28
So
y=6e^(4x-28)
Check:
y'(x)=dy/dx=24e^(4x-28)
y'(2.5)=3.655*10^-7
y(2.5)=9.138*10^-8
y'(2.5)/y(2.5)=4.000 ok