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Is this an homogeneous equation?

  1. Oct 22, 2012 #1
    Actually I can't find if a differential equation is homogeneous or not

    I thought homogeneous is given by

    dy/dx= f(x,y)/ g(x,y)

    but it doesn't look like that


    For eg:

    dy/dx= (y+x-1)/(y-x+2) is not homogeneous at all though

    f(x,y)=y+x-1 and g(x,y)=y-x+2

    How can you tell that if an equation is homogeneous or not?

    Thanks a lot :)
     
  2. jcsd
  3. Oct 22, 2012 #2
    Hello.

    Well, your definition is not right. [itex]y'=f(x,y)[/itex] is homogeneous if [itex]f(tx,ty)=f(x,y)[/itex] for any nonzero t. You can use that as a test. Basically, if you substitute tx and ty for x and y, then the t should divide itself out.

    Example: [itex]y'=\frac{x^2+xy}{\sqrt{x^4+y^3x}} [/itex].
    [itex]\frac{(tx)^2+(tx)(ty)}{\sqrt{(tx)^4+(ty)^3(tx)}}= \frac{t^2(x^2+xy)}{\sqrt{t^4(x^4+y^3x)}}=\frac{x^2+xy}{\sqrt{x^4+y^3x}}.[/itex]
     
    Last edited: Oct 22, 2012
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