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Can we use y=vx for non-homogeneous differential equation?

  1. Mar 16, 2013 #1
    1. The problem statement, all variables and given/known data
    Can we use y=vx for non-homogeneous differential equation?

    Example:
    yy'=x^3+(y^2/x)→not homogeneous


    2. Relevant equations
    y=vx
    dy/dx=v+x(dv/dx)

    3. The attempt at a solution
    By substituting the equation above:
    vx(v+x dv/dx)=x^3+(v^2 x^2)/x
    v^2*x+vx^2 dv/dx=x^3+v^2*x
    Eliminate the v^2*x:
    vx^2 dv/dx=x^3
    Divide both sides with x^2:
    v dv/dx=x
    vdv=xdx
    Continue the integration:
    y^2=x^2(x^2+c), where c is a constant
     
  2. jcsd
  3. Mar 16, 2013 #2
    The substitution y=vx makes it easier to find the solution, because a homogenous differential equation takes the form:

    ##\frac{dy}{dx}=\frac{f_1(x,y)}{f_2(x,y)}##

    which then equals ##\frac{f(y/x)}{g(y/x)}## or ##\frac{f(x/y)}{g(x/y)}## by taking ##x^n## or ##y^n## common, if f1 and f2 are homogenous in degree n.
     
  4. Mar 16, 2013 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    Easier: ##v \, dv/dx = d(v^2/2)/dx.## And, there are two solutions.
     
  5. Mar 17, 2013 #4
    May I know the hint to the other solution? Thanks
     
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