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Substitution to convert first order ODE to homogenous

  1. Apr 3, 2015 #1
    1. The problem statement, all variables and given/known data
    Use the substitution ##x=X+h## and ##y=Y+k## to transform the equation
    ##\frac{dy}{dx}=\frac{2x+y-3}{x-2y+1}## to the homogenous equation
    ##\frac{dY}{dX}=\frac{2X+Y}{X-2Y}##
    Find h and k and then solve the given equation

    2. Relevant equations


    3. The attempt at a solution
    If I simply make the substitution into the equation, I get a homogenous equation which I can solve using y=vx substitution. But what I need help understanding is how the ##\frac{dy}{dx}## becomes ##\frac{dY}{dX}## after simply substituting into the LHS?
    Is some proof or method of doing this so that I can turn dy/dx into dY/dX and vice versa? The chain rule doesn't help, as I cannot relate X and Y
     
  2. jcsd
  3. Apr 3, 2015 #2

    HallsofIvy

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    Staff Emeritus
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    First [tex]\frac{dY}{dx}= \frac{d(y- k)}{dx}= \frac{dy}{dx}[/tex]

    Now use the chain rule. [tex]\frac{dY}{dX}= \frac{dY}{dx}\frac{dx}{dX}= \frac{dy}{dx}(1)[/tex]

    The real point is that both x= X+ h and y= Y+ k are linear with slope 1.
     
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