Conventional solution of Poission’s equation involve solution

This is a two series solution which is tedious to solve.

The book PDE by Asmar suggested a method of solving Poisson problem with one series by lumping function of y into the coefficient by using:

.AND

The book gave the final equation of:

where the associate homogeneous solution is:

This book claimed this is by using variation of parameters using h1 and h2 as y1 and y2 obtain from solving the associate homogeneous equation.

I use the standard variation of parameter and cannot get the same answer. Can someone point me to a web site to verify the book? I hate to say the book is wrong but I did triple verified and fail. I have not manage to find anything on this from 4 other text book nor on the web to even talk about single series solution.

Is it really important to use single series rather than two series because I have not problem doing in the convensional way using two series, it is very easy to understand. It is the book trying to be simple and jump steps that I don't agree with their formula all all.

I think in should not be integrated like this in the definite integral. Because it really a function of y, not a function of (b-y)

I think the solution using variation of parameters should be:

.

Please tell me whether I am correct, no more of the dummy variable "s".

Thanks