Question

Suppose Y1 is a function of x which dy1/dx=3y1. Suppose y2 is a function of x which dy2/dx=8x+5. If the graphs of y1 and y2 have the same y-intercept and they intersect at x=2, then determine the value of the y-intercept.

Answer 1

if dy1/dx = 3y1
then y1 = ae^(3x) , where a is a constant, a≠ 0

(check: dy1/dx = 3a(e^(3x)) = 3y1 )

if dy2/dx = 8x + 5
then y2 = 4x^2 + 5x + c

the y-intercept of y1 is (0,a), but that is also the y-intercept of y2

so in y2:
a = 0 +0 + c
a = c

when x=2
y1 = a(e^6) , y2 = 16+10+c = c + 26

but a=c, so
a(e^6) = a+26
a(e^6) - a = 26
a(e^6 - 1) = 26

a = 26/(e^6 - 1) , which would be the y-intercept