d is meant to be partial derivatives.
Say we want to solve this subject to some curve with .
Use the chain rule,
By the method of charachteristics we set,
The solution to the IVP problems are:
The solution to #3 is simple:
This is a Seperable equation,
Where is an arbitrary starting point on the solution curve.
By (*) we see that .
Thus by (**),
Where is any arbitrary differencial function (and not equal to ).
Which seems to work !!! Because if everywhere then . If you substitute that into the PDE you get: