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Differential equations problem

  1. Mar 28, 2010 #1
    1. The problem statement, all variables and given/known data
    A curve with Cartesian equation y=f(x) passes through the origin. Lines drawn parallel to the coordiante axes through and arbitrary poing of the curve form a rectangle with two sides on the axes. The curve divides every such rectangle into two regions A and B, one of which has and area equal to n times the other. Find all such functions f.

    2. Relevant equations

    3. The attempt at a solution

    Obviously f(x)=cx where c is a real number works, but there must be others. Any ideas?
  2. jcsd
  3. Mar 29, 2010 #2
    [tex]\int[/tex]ydx (from 0 to x) = kxy ( for k = +- 1/n+1 or n/n+1).
    This gives y = k(x y' + y) ; hence y =const. x^(1-k)
  4. Mar 29, 2010 #3


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    The area below the curve is, of course, [itex]\int_0^{x} f(t)dt[/itex] while the area above the curve is [itex]\int_0^{x} f(x)- f(t) dt= xf(x)- \int_0^{x} f(t) dt[/itex] Saying one is n times the other means that [itex]n\int_0^{x}f(t)dt= xf(x)- \int_0^x f(t)dt[/itex] so [itex]xf(x)= (n+1)\int_0^x f(t) dt[/itex].

    Differentiating both sides of that with respect to x will give you a differential equation for f(x)- xdf/dx+ f(x)= (n+1)f(x) so that xdf/dx= n f(x). That is separable and easily integrable.
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