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

[artan]

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

 

[JJacquelin]

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).

 

[artan]

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

 

[artan]

Can you write the special function and show me some steps how to find particular solution.Thank you

 

[kosovtsov]

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.

 

[JJacquelin]

The special function is Ei(X)
http://mathworld.wolfram.com/ExponentialIntegral.html
As already said, let y(x)=x*f(x) which leads to
x f '''+(2-x)f ''= x +(3/x²)
Ei(-x) will appear in solving this equation.

 

[artan]

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 don't know how to find it
thank you.

 

[JJacquelin]

I started solving this DE this way:
y=x^m

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.

 

[artan]

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.

 

[artan]

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.

 

[JJacquelin]

Hi !

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

 

[artan]

Thank you for the help