du/dt-t(du/dx)=-u^2
d is meant to be partial derivatives.
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 is any arbitrary 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.