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Homework Help: How to find critical points for a differential equation?

  1. Sep 27, 2013 #1
    1. The problem statement, all variables and given/known data

    f(x,y)= 3x5y2 - 30x3y2 + 60xy2 +150

    How do I find the critical points where this equation equals zero?

    2. Relevant equations


    3. The attempt at a solution

    I found the partial derivatives, but I'm stuck now and don't know where to go from here.

    dx = 15x4y2 - 90x2y2 + 60y2
    dy = 6x5 - 60x3y + 10xy + 240
  2. jcsd
  3. Sep 27, 2013 #2


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    start by setting both equal to 0! then try quadratic factoring for the dx. the dy should be obvious.
  4. Sep 27, 2013 #3


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    also, you ask for critical points and where the equation equals zero...are you sure both occur simultaneously? i havent checked but dont assume this to be the case
  5. Sep 28, 2013 #4
    Okay so I did that. I got for dx, x = 3 +- sqrt(20)/2
    The first part of dy should actually be 6yx^5.

    After I have the dx value, I can sub them into the dy equation which gives approximately y = -0.1545 and y = -3.6099. Where do I go from here? There should be 4 separate (x,y) points.
  6. Sep 28, 2013 #5


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    Spell things out a little more carefully. What you quote a solution for I think is really saying

    ∂f/∂x = 0 at all points along (x/y)2 = those numbers, which is actually four lines through the origin I think.

    Maybe there is a nice algebraic breakup of your ∂f/∂y but anyway what you are doing should lead you to the four points along the four lines.

    Alternative approach is to find (dy/dx)f and its reciprocal and find the points where both are 0. Well same thing really, -(dy/dx)f is the ratio of your two derived polynomials.

    Also I think the second of yours, what you call "dy" is mistaken and it works out fairly simply.

    You might as well have taken the factor 3 out of your starting equation.
    Last edited: Sep 28, 2013
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