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Please if anyone can help me to solve this differential equation.

  1. Mar 24, 2012 #1
    Please if anyone can help me to solve this differential equation.
     
  2. jcsd
  3. Mar 25, 2012 #2
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    Hello,

    Two obvious solution of the homogeneous part of the equation are y=x and y=x². The third is a special function.
    Nevertheless, particular solutions for the whole ODE can be derived :
    Let y=x*f(x) and solve the ODE which unknown is f(x).
     
  4. Mar 25, 2012 #3
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    I can find that first and second obvious solution of the homogeneous part x and x^2,but how can I find the third,it is something of x^x.


    Can anyone explain how to find the particular solution showing me some steps of the solution.Thank you
     
  5. Mar 25, 2012 #4
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    Can you write the special function and show me some steps how to find particular solution.Thank you
     
  6. Mar 25, 2012 #5
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    The general solution to your ODE is as follows

    [itex]y(x) = -3x(x-3)\ln(x)+[(-\frac{1}{2}x^2+x)\int_{-x}^∞ \frac{\exp(-t)}{t}dt -\frac{1}{2}\exp(x)(x-1)]C_1-\frac{x^3+9}{2}+x^2C_2+xC_3[/itex]

    where [itex]C_i[/itex] are arbitrary constants.
     
  7. Mar 25, 2012 #6
  8. Mar 25, 2012 #7
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    I started solving this DE this way:
    y=x^m
    y'=m*x^(m-1)
    y''=m(m-1)x^(m-2)
    y'''=m(m-1)(m-2)x^(m-3)

    and replace them in DE we get:

    (m-1)(m-2)(m*x^(m-1)-x^m)=0
    so we get :
    y1=x
    y2=x^2
    m*x^(m-1)-x^m=0 this is the third solution but i dont know how to find it
    thank you.
     
  9. Mar 25, 2012 #8
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    You suppose that the third solution is on the patern y=x^m which is not the case. As a consequence this method cannot lead to the third solution.
    By chance, the first and the second solution are on the patern y=x^m, so leading to m=1 and m=2. But obviously not the third.
     
  10. Mar 25, 2012 #9
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    MR.Kosovtsov can you write the whole method how did you get the solution,not just the final solution.
    I will be grateful.
    Thank you.
     
    Last edited: Mar 25, 2012
  11. Mar 25, 2012 #10
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    which method should i use to get the solution.
    Can you write for me the beginig of the method you use?
    I will be grateful.
    Thank you for the help.
     
    Last edited: Mar 25, 2012
  12. Mar 25, 2012 #11
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    Hi !

    In attachment, the solution for the homogeneous ODE.
    Same method for the complete ODE.
     

    Attached Files:

  13. Mar 25, 2012 #12
    Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

    Thank you for the help
     
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