Question

PDE help, please
if v,w are solution of \(u_t+uu_x+u_{xxx}=0\)
show that v+w is also a solution only when v*w does not depend on x

Answer 1

\((v+w)_t+(v+w)(v+w)_x+(v+w)_{xxx}=0\)
if and only if \(vw_x+wv_x=0\)
but I don't know how to turn it to vw does not depend on x

Answer 2

@wio

Answer 3

v*w does not depend on x is the same as:

\[(v*w)_x = 0\]

Answer 4

\[
vw_x+wv_x = (vw)_x = 0
\]

Answer 5

This is the product rule

Answer 6

What?? that is ... amazing!! OMG.Thank you so much. I thought of something on the sky!!