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I If y is a function of x, then x is also function of y?

  1. Aug 21, 2016 #1
    So then if you take the integral of y(x) dy = x dy, you get y^2 = xy.

    But if x is a function of y, that means the equation is y(x)dy = x(y) dy, which gives you y^2 = X(y), where X is the integral of of x(y) dy?
     
  2. jcsd
  3. Aug 21, 2016 #2
    if ##y=f(x)##, x is a function of y, i.e. ##x=f^{-1}(y)##, only if ##f## is invertible. Look up inverse function theorem, for what you need.
     
  4. Aug 21, 2016 #3

    Mark44

    Staff: Mentor

    This appears to be an intermediate step in solving a differential equation.

    Starting with y dy = x dy, if you integrate the left side you get y2 + a constant, but you can't integrate the right side. Is x dy a typo? Should it be x dx? If so, and we start with y dy = x dx, integrating both sides yields y2 = x2 + C.
     
  5. Aug 21, 2016 #4

    Mark44

    Staff: Mentor

    The short answer to your thread title -- "If y is a function of x, then x is also function of y?" -- is "Not necessarily."

    For example, if y = f(x) = x2, then y is a function of x, but x is not a function of y. As already stated, each variable is a function of the other only if both functions are one-to-one.
     
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