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Partial Differential/Integration Arbitrary Functions

  1. Feb 23, 2014 #1
    Use integration to find a solution involving one or more arbirary functions
    [tex]\frac{\partial u}{\partial y}=\frac{x}{\sqrt{1+y^2}}[/tex]
    for a function [itex]u(x,y,z)[/itex]
    [tex]u(x,y,z)=x\int \frac{dy}{\sqrt{1+y^2}}[/tex]
    let [itex]y=\sinh v[/itex]
    [tex]u(x,y,z)=x\int \frac{\cosh v\: dv}{\sqrt{1+\sinh ^2v}}[/tex]
    [tex]u(x,y,z)=x\sinh ^{-1}y+f(x,z)[/tex]
    So here's the question. Why is the solution with an arbirary function [itex]f(x,z)[/itex] and not two arbitrary functions [itex]f(x)+g(z)[/itex]? What's the difference?
     
  2. jcsd
  3. Feb 23, 2014 #2

    AlephZero

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    ##f(x,z)## is more general than ##f(x) + g(z)##.

    For example, how would you convert a function like ##f(x,z) = x^z## to the form ##f(x) + g(z)##?
     
  4. Feb 23, 2014 #3
    Good point. But how can i explain that in a mathematical language?
     
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