To be honest, I'm confused about what's confusing you, so I don't know how much help I can be
Let's consider a set of 15 ordered pairs:
(0,2) (1,2) (2,2)
(0,1) (1,1) (2,1)
(0,0) (1,0) (2,0)
(0,-1) (1,-1) (2, -1)
(0,-2) (1,-2) (2,-2)
Given y'=slope=2y, we note the slope at these points:
4 4 4
2 2 2
0 0 0
-2 -2 -2
-4 -4 -4
(Poorly) graphically represented, with slopes (4,2,0,-2,-4) given (/,↗,-, ↘,\):
Code:
2 / / /
1 ↗ ↗ ↗
0 - - -
-1 ↘ ↘ ↘
-2 \ \ \
0 1 2
Over more points, we see trends comparable to the graphs of y=1*e^2x, y=0*e^2x, and y=-1*e^2x. In fact the general solution to y'=2y is y=C*e^2x.
Hope this helps...