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PDE: Solving to find a constant c

  1. Jan 24, 2016 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    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 ?

    2. Relevant equations

    N/a
    3. 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.
     
  2. jcsd
  3. Jan 24, 2016 #2
    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.
     
  4. Jan 24, 2016 #3

    RJLiberator

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    Gold Member

    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?
     
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