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Maxima and minima of differential equation

  1. Sep 26, 2014 #1
    1. The problem statement, all variables and given/known data
    Consider the differential equation [itex]y'=x-y^2[/itex]. Find maxima, minima and critical points; show that for every solution [itex]f=f(t)[/itex] exists [itex]T\geq 0[/itex] such that [itex]f(t)< \sqrt{T}\;\forall t > T[/itex]

    2. Relevant equations

    The Riccati equation: [itex]y'=a(x)y^2+b(x)y+c(x)[/itex]
    The Bernoulli equation: [itex]y'=a(x)^n+b(x)y[/itex]

    3. The attempt at a solution

    I've been trying to study the differential equation given by [itex]y'=x-y^2[/itex] for a while, and I didn't was even close to a solution. Finally when I gave up I solved this equation with Wolfram Alpha, and it seems that the solution was quite intricate and doesn't seem possible to solve only with elementary functions, see Wolfram's solution.

    Now my question is, what are the possible methods to use when I encounter a non-linear equation of like this?, is it possible to say something about it without the need of a computer?.

    One thing I saw is that this is similar the Riccati equation [itex]y'=a(x)y^2+b(x)y+c(x)[/itex] using [itex]c(x)=x, a(x)=-1, b(x)=0[/itex]
    and is possible to get a general solution if I know two other solutions: if [itex]f_1,f_2[/itex] are solutions then [itex]f=f_1+C(f_2-f_1)[/itex] is a general solution. The problem here is that I'd need to find particular solutions for the equations, something that I couldn't figure out how to do it in the general case.
  2. jcsd
  3. Sep 27, 2014 #2


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    I would suggest that you try to answer the questions without trying to solve the DE itself. I haven't worked it all the way through myself, but you can certainly find the points where a solution might have ##y'=0## and you can also figure out the sign of the second derivative at such points without solving the DE, so you can answer the critical point questions.

    I'm not sure about that second part, but I wonder if your text has been discussing ways to examine the equation qualitatively without solving it. That's where I would look for possible techniques.
  4. Sep 28, 2014 #3


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    @trash: So have you given this any thought? Or have you just abandoned the thread?
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