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Struggling through the early chapters of partial differential equations (from the all-time-classic Stepanov's book) I often run across the moves that are not really technically demanding, but the logic behind takes time to sink in.

One of them is the trick used to find a particular solution of the first order linear PDE satisfying initial condition:

Here is an excercise: find the solution of the following equation

         satisfying the condition: when

We start by writing out the associated system of ODEs:

Use the first identity to find a "first integral" of the system:

Then equate the first and the third quotients to obtain the second "first integral":

Obviously there are no more independent first integrals

According to the theory, the general solution is given as an arbitrary function of these two integrals:

Now we shall find the particular solution. Let x=1. Following the book, we introduce new functions and to which and turn when we set the value of :

Now we solve these equations with respect to and :

And now comes the difficult part. In order to get the final result we need to substitute the above results in the expression of replacing with :

It took me some time to accept that I have to substitute which is a more general expression than  in the equation which was derived after giving ist particular value.

Finally:

The reasoning is clear, once it is explained.  and are both solutions, hence the function of them is also a solution. Then, if we let they turn to  and . Then by virtue of the equations (1) we get the variables and and consequently the expression of as required.

In the book the whole argument is presented in the most general form. However, it takes several excercises and hours of thinking to get a hands-on experience with this method.