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Sketch the graphs of solutions of diff. eqs

  1. Oct 3, 2011 #1
    Hi people!
    This is my first topic here so excuse me if im doing smth wrong)

    So basically im having problems with understanding of how to sketch
    the graphs of solutions of diff. eqs in terms of y and t...

    Here is the description and 2 problems:
    Problems 8 through 13 involve equations of the form dy/dt = f (y). In each problem sketch
    the graph of f (y) versus y, determine the critical (equilibrium) points, and classify each one as
    asymptotically stable, unstable, or semistable (see Problem 7).

    dy/dt = y(1 − y2), −∞ < y0 < ∞ (1)

    dy/dt = y2(1 − y)2, −∞ < y0 < ∞ (2)

    so for (1) i have the following:
    f(y)=-y3+y and the roots are f(y)=0 then y=0,1,-1.
    so for each interval we have:
    −∞,-1 (+)
    -1,0 (-)
    0,1 (+)
    1,∞ (-)
    which means that -1 and 1 are stable crit. points
    and 0 is unstable
    I have a cubic parabola for f(y),y plane and i dont know how to interpret all this
    data in terms of y,t plane

    for (2)
    0 and 1 are both semi stable points since the function doesn't change its sign..
    same story,cant move to t,y=(
    Please help me to understand the idea here!
    would be great to see ur sketches=)
    Thank u all!
     
  2. jcsd
  3. Oct 3, 2011 #2
    You need to write the equations more precicely like for the second one:

    [tex]\frac{dy}{dt}=y^2(1-y)^2[/tex]

    Ok, so you find the critical points, where the derivative is zero. So for that one, they are 0 and 1. Now, what I suggest you do is code the slope fields in some language. Below I use Mathematica to draw the slope field as well as a test case (in red) with y(0)=0.5. Notice how the test case follows the slope field:

    Code (Text):

    mysol = NDSolve[{y'[t] == y[t]^2 (1 - y[t])^2, y[0] == 0.5},
      y, {t, 0, 5}]
    p1 = Plot[y[t] /. mysol, {t, 0, 5},
      PlotStyle -> {Thickness[0.008], Red}]

    Show[{StreamPlot[{1, y^2 (1 - y)^2}, {t, 0, 10}, {y, -5, 5}], p1}]
     
    Notice how initial points between 0 and 1 asymptotically approach the equilibrium point at 1. Notice how initial conditions outside of 0 and 1 move to or away from the equilibrium points. Try and work with this code or some other code in matlab or whatever and learn how to interpret the slope fields in terms of the stability of the equlibrium points.
     

    Attached Files:

  4. Oct 3, 2011 #3
    Thank you for the tips with Mathematica jackmell, im sure going to need it! But the problem is that i have to do it by hand... Its not a computer based course, so if i encounter one of these on the exam, i wont be able to use Mathematica..
    I know that the basic idea is that the solution graphs approach the stable equilib. diverge from the unstable, and approach the semi stable points on one side (depends on arrows), but the graphs (of sols) have inflection points, they may go in different directions beyond the given points...etc
    So how can i at least approximately sketch it by hand??
     
  5. Oct 4, 2011 #4
    Ok, sorry but I'm pressed for time to show you that. However, "Differential Equations" by Blanchard, Devaney, and Hall does a good job of constructing slope fields qualitatively by analyzing the function f(y). Try and find that book in the library.
     
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