Let's start with the homogeneous equation...

(1)

... one solution of which we know to be . If u(x) and v(x) are solutions of (1) then...

(2)

Multipling the first equation by v, the second by u an taking the difference we obtain...

(3)

... that with the substitution becomes...

(4)

The (4) is easily solved and gives...

(5)

where c is a constant that can be set =1 without losing anything. Deviding both terms of (5) by we obtain the new equation...

(6)

Since is solution we can set in (6) obtaining...

(7)

... the solution of which is...

(8)

The general solution of (1) is then...

(9)

Let's consider now the 'complete' equation...

(10)

With a little of patience you find that is a particular solution so that the general solution of (10) is...

(11)

Kind regards