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

1. Mar 24, 2012

artan

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

2. Mar 25, 2012

JJacquelin

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).

3. Mar 25, 2012

artan

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

4. Mar 25, 2012

artan

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

5. Mar 25, 2012

kosovtsov

Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

The general solution to your ODE is as follows

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

where $C_i$ are arbitrary constants.

6. Mar 25, 2012

JJacquelin

7. Mar 25, 2012

artan

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.

8. Mar 25, 2012

JJacquelin

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.

9. Mar 25, 2012

artan

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
10. Mar 25, 2012

artan

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
11. Mar 25, 2012

JJacquelin

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:

• Homogeneous ODE.JPG
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12. Mar 25, 2012

artan

Re: need:x^2y'''-x^2y''+2xy'-2y=x^3+3x

Thank you for the help

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