# 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