how solve this differential equation?

[ghasem_008]

Hi
how solve this differential equation?
x²y'' + xy' = 1/(1-x) 0<x<1 ; y(0.5)=3

my problem is in inhomogenous part.

I know that it's homogeneous solution is :
y = - Ln(x)

how get its' inhomogeneous solution?
thanks

 

[albbg]

I think you should apply D'Alambert method to find a linear independent solution (from that you already found) of the homogeneous.
Then when you'll have y1h=-ln(x) and y2h=found using D'Alambert you can apply the Lagrange method to find the complete solution.

 

[_Eduardo_]

Hi
how solve this differential equation?
x²y'' + xy' = 1/(1-x) 0<x<1 ; y(0.5)=3

First solve x²u' + xu = 1/(1-x) -------> u(x) = (C - log(1-x) + log(x))/x Then integrate u(x).

Unfortunately, integral of log(1-x)/x can't be expressed by simple functions.


https://www.wolframalpha.com/input/?i=solve+x²y''+++xy'+=+1/(1-x)
Thank's WolframAlpha!

 

[zorro]

First solve x²u' + xu = 1/(1-x)

x(xu)' = 1/(1-x)
(xu)' = 1/[(1-x)x] = 1/x + 1/(1-x)
---> Eduardo's result

Z

 

[ghasem_008]

thank you very much fro your attentions...
So,I can not obtain a analytically solution?

 

[phongphanp]

Thing may be need the advance Laplace transform in this problem.

 

[ghasem_008]

thank you for reply
how use from advance Laplace transform in this problem?can you explain it,briefly?

 

[phongphanp]

Like curvelet study in Nonlinear science and Chaostics theory.

 

[ghasem_008]

how can I solve this equation by your suggestion?

- - - Updated - - -

how can I solve this equation by your suggestion?

 

[phongphanp]

Sorry, It's have copyright but if you try from basic,it's may be advanced. Try computer programming generate solution. The number of control point may in form 2N , I'm think should try sample by niquist rate .

 

[phongphanp]

I think you should read the book ,"Differential Equations". And other book on science.