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Math Help - [SOLVED] Slope Field

  1. #1
    Member ronaldo_07's Avatar
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    [SOLVED] Slope Field

    Sketch the direction field of the differential equation

    y′ = 2 − 3y + y^2

    in the domain x \epsilon[0, 3], y \epsilon[−3, 3].

    Using the direction field decide whether a solution y(x) tends to a finite limit when x\rightarrow\infty.

    How does this limit depend on the value of the initial condition y(0) ?
    ------------------------------------------------------------------------

    Im not sure how to get the values to draw this directional field. Do I just sub in y=-3.-2.-1.0.1.2.3 ? and why do I have the domain of x when there is no x in the equation?
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  2. #2
    MHF Contributor

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    Quote Originally Posted by ronaldo_07 View Post
    Sketch the direction field of the differential equation

    y′ = 2 − 3y + y^2

    in the domain x \epsilon[0, 3], y \epsilon[−3, 3].

    Using the direction field decide whether a solution y(x) tends to a finite limit when x\rightarrow\infty.

    How does this limit depend on the value of the initial condition y(0) ?
    ------------------------------------------------------------------------

    Im not sure how to get the values to draw this directional field. Do I just sub in y=-3.-2.-1.0.1.2.3 ? and why do I have the domain of x when there is no x in the equation?
    Someone else already asked the same problem, but said (s)he could sketch the direction field so I'll answer to you anyway.

    At the point (x,y), the slope is 2-3y+y^2. It doesn't depend on x: it means that the slopes will be the same on any "column". The direction field is invariant by translation along the x-axis. So you have to sketch one column of slopes (for instance at x=0, or anywhere else), and copy it for a few other values of x.

    How to draw the directions? The main point is to see where the slope is positive or negative, or zero. Thus, as a start, you should study the sign of 2-3y+y^2. Then draw the horizontal slopes (those equal to 0). You can draw the maximum/minimum slope if any, and add a few slopes to fill your sketch in and get a better feeling of what the field looks like.
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