Lets say for example, we are given:
dy/dx - 4y = 2 or y' - 4y = 2 , y(0) = 4
=> M= e^(-4t)
e^(-4t) y' - 4e^(-4t)y = 2 e^(-4t)
e^(-4t) y = -1/2 [ e^-4t ] + C
y = -1/2 + Ce^4t
When t = 0, y = 4
4 = -1/2 + C
C = 4.5
therefore... y = -1/2 + 4.5e^4t
Now, is -1/2 the null function or particular? what about for 4.5e^4t?
I was imagine a long period of time before the "system" starts, so t = negative infinity, then the exponential function would approach 0, and we would be left with y = -1/2
if t = 0, when the system just starts, we were given y = 4
now after a very long period of time, t = infinity
y = infinity
What does this mean? the system has no steady state?