Given the linear second order 'incomplete' DE...

(1)

... we are able to solve it from the knowledge of two independent solutions u and v. Because u and v are solutions of (1) You can write...

(2)

... so that subtracting with some easy step You obtain...

(3)

Now setting the (3) becomes...

(4)

... the solution of which is very easy...

(5)

Deviding both terms of (3) by and taking into account (5) You obtain...

(6)

Now it is easy to verify that is solution of (1) so that the general solution of (1) is...

(7)

The integral is (7) is a non elementary function known as 'exponential integral function'...

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