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Cauchy-Euler Differential Equation

  1. Jul 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the equation x2y''-2xy'+2y=x with bountary conditions y(1)=0, y(2)=0.

    I don't know how to solve this without treating it as a Cauchy-Euler equation but I'm struggling because the equation equals x.


    3. The attempt at a solution
    By treating this as a Cauchy-Euler equation x2y''-2xy'+2y=0 and using y=xr, I get that r=2,1.
    I can't find a particular integral for the equation though and I'm not even sure this is a Cauchy-Euler equation anymore.
     
  2. jcsd
  3. Jul 23, 2012 #2

    HallsofIvy

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    Yes that is a 'Cauchy-Euler' equation and, yes, its characteristic equation is r(r-1)- 2r+ 2=(r- 2)(r- 1)= 0 so that x and [itex]x^2[/itex] are solutions. Since "x" is already a solution, try y= Ax ln(x) as a solution to the entire equation.

    That works (and Cauchy-Euler equations are especially easy) because the change of variable [itex]x= e^t[/itex] reduces that equation to the equation with constant coefficients [itex]y''- 3y'+ 2y= e^t[/itex] (the primes now indicate differentiation with respect to t).
     
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