PDE: Solving to find a constant c

[RJLiberator]

Homework Statement

Consider the nonlinear (ordinary) differential equation u' = u(1-u).
a) Show that u_1 (x) = e^x/(1+e^x) and u_2(x) = 1 are solutions.
b) Show that u_1+u_2 is not a solution.
c) For which values of c is cu_1 a solution? How about cu_2 ?

Homework Equations

N/a

The Attempt at a Solution

a) To show that they are a solution I plugged in the corresponding u_1 and u_2 into the equations and proved equality.
b) To show that this is not a solution, I did the same.
c) This is where I am having trouble, here is my strategy:

We observe c*u_2 first.
we know u_2' = 0 so from u'=u(1-u)
we have c*u' = 0 and we know that 1*c = c so on the right hand side we have c(1-c) and so we have:
c-c^2 =0
This means c is either 0 or 1.

This doesn't seem right, there seems to be a better way (perhaps involving differential equations knowledge) to solve for c. I would imagine c could be in the form e^x of some sort.

Similarly with c*u_1 I am finding equally difficult problems where c = 0.

 

[Dr. Courtney]

Substitute in the proposed solution.

Apply all the operations and simplify.

See if you can find a value of C for which the resulting equation is true.

 

[RJLiberator]

So, when I solved for part c for u_2 and found that c=c^2
This means that c = 0 or 1.
That would be a correct solution then?