I am going to try. I cannot gaurentee my solution is correct.

Say we want to solve this subject to some curve with .

Then

Use the chain rule,

By the method of charachteristics we set,

1)

2)

3)

The solution to the IVP problems are:

1) (**)

2) (*)

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 isanyarbitrary differencial function (and not equal to ).

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Which seems to work !!! Because if everywhere then . If you substitute that into the PDE you get:

Which works.