Thread: PDE-Characteristic Curves

Let be the soloution of that is defined in the whole plane. Prove that .
Hint: Think of the characteristic curves of this PDE.

HOPE You'll be able to help me
Thanks in advance!

Originally Posted by WannaBe

Let be the soloution of that is defined in the whole plane. Prove that .
Hint: Think of the characteristic curves of this PDE.

HOPE You'll be able to help me
Thanks in advance!

Solve the characteristic system and assure that any nonzero solution on the whole plane will develop a discontinuity at the origin.

I've already done this...We'll get:
and: where is k is a function of y ...
We'll have a discontinuity every time that and every time that x=0...
How did you deduced the concloution that the discontinuities are at the origin?

Thanks !

From y=c_1x. Thus there will always be a discontinuity at the origin, but the expression lnx+k(y) may be nonzero for a suitable choice of initial data.

is a line segmant in ...Why there is a discontinuity at the origin?

It is one of the characteristics though. The constant c_1 is the slope and will vary along each characteristic, while being (explicitly) determined by c_1=y/x. So at x=0 (for which y=0) the slope will become infinite and the solution will develop a shock. check your notes.

actually, at x=0, the expression ln(x) has a discontinuity... So we can deduce the result from this (I think...)...

Thanks...I think I understand it