@Kitonum Thank you a lot for your support.
Is not exacly what I wanted. 
The aim is to replicate the same field vector I got aroung phi direction, at the end it will look like a disk. This is to evaluate each vector in r and phi direction, on theta=pi/2

@sand15 That really helped me, to do the tayor series in the k function. Thank you a lot for your answer, the only I can say is that this is a great community always willing to help.

Here upload the file ended with the advices that you both told me. 

prueba_coeficientes.mw

@acer Really appreciate your soon answer, It was what I wanted, thank you so much friend! 

@dharr Thanks a lof for your advice. I did what you mentioned, rescaled all the constants, and change the variable for a dimensionless one, did it works for the radial part. Best regards friend

@dharr Hello dharr, I want to tell you thanks again for all your help, had a lot of significanse for me, and also if can help me again, would want to ask you something else.
I have been working on the equation using different values for the constants; all of them are about physics and the equation represents the wave equation under the influence of a black hole. 
At the time to use greather values for "r" the progam showed the error: "Warning, cannot evaluate the solution further right of .99675383+0.*I, maxfun limit exceeded". Then I used the code maxfun=0 and stepsize = 1000  to see the plot around r=12*10^6 because there is the event horizon of the black hole. The question is if am I using the correct comand? and also on the sencond part is not letting me plot the equation.

MODIFIED_DIFFERENTIAL_EQUATION_2.mw

@dharr You have done a great job, jus what I was looking for. The value choiced is the better. Now I can see the right way to solve this kind of equations. Really thanks you for your help, 

@acer 

Thanks also for the explanation about the mistake in the integrand, a great point to take into account.

This day learnt a lot from you, and hope to replícate This in to my next assigments. 

Have a good day @acer, you are a great man!

greetings from Mexico

@acer 

wow man, I'm really impressed with your answer, is correct and you respond almos instantly. Thank you for you dedication, your time and atention with me. I desire some day got as much knowledge as you have. 
God bless you!!!!!!!

@acer really thanks for your kind answer. As you show is posibly to plot the sequence. Due to your great domain in the subject, I also would like to ask you something else: why  the graph is diferent from the one on matlab? If the code is almost the same.

We both started obtaining the valúes for lambda  that optimize the equation:

-2/3 + int(sqrt(x)/(exp((x-lambda)/tau ) + 1), x=0..infinity);

varying tau in the loop 

@mmcdara I am really thanks for your help.

Best regards friend 

@Axel Vogt thank you so mucho for your support, was really usefull your answer and helpped me to solve a problem in statistical physics.

Best regards 

@Preben Alsholm thanks a lot, is just what i did

N:=5

phi:= Matrix(10, N, (x,n) -> sin(n*x)); 

best regards 

@tomleslie thank you, your advice helped me, just use the code 

N:=5

phi:= Matrix(10, N, (x,n) -> sin(n*x)); 

best regards