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I Set y/x as y=ux to solve ode

  1. Mar 10, 2016 #1
    I am studying ode now, and my text has that
    If y'=f(y/x)
    Then, setting y/x=u ; y=ux is a way to solve it.
    I understand the idea, turn orignal form to separable form.

    But I can't get the differentiation, Book says
    y'=u'x+u by product rule which I already know.
    Here my question is why u=y/x that obviously has two variables x & y, u(x,y) should be differentiated respect to x like it only has one variable x ( like u(x) )
  2. jcsd
  3. Mar 10, 2016 #2


    Staff: Mentor

    x is being differentiated.
    Starting with y = ux, we differentiate everything with respect to x.
    y' = ux' + u'x
    Here, x' means ##\frac{d}{dx}x##, which simplifies to 1, leaving us with ##y' = u \cdot 1 + u'x = u + u'x##.
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