Investigating the Behavior of a Diff. Eq. w/ Constant K & W

[David Donald]

So I have a diff eq
dy/dx = ky - w

where K and W are constant's
I want to draw a phase portrait
so i set ky - w = 0

and determine the expression equals 0 at y = w/k

so I want to study the behavior of the solution greater w/k
and less than w/k
so I plug (2w/k) and (-2w/k) and check the sign

so k(2w/k) - w = w Positive

and

k(-2w/k) - w = -3w Negative

Did I determine this correctly? I feel like I am doing something horribly wrong here

 

[Ssnow]

you must specify the sign of the constants $k,w$ before, so you know where put the constant solutions on the phase plane ...

 

[S.G. Janssens]

Did I determine this correctly? I feel like I am doing something horribly wrong here

Yes, assuming your constants $k$ and $w$ are positive, also see the remark by @Ssnow. Is this a simple model of an exponentially growing population with growth rate $k$ and constant harvesting $w$?

Two more remarks:

1. You have a first order equation, so there isn't really a phase plane or a phase portrait but more a phase line.
2. You probably already knew this, but you can easily solve the ODE explicitly. However, a graphical analysis like yours also gives all the information.