Question

Find the general solution of the partial differential equation Ux = 0, where the unknown function u := u(x,y), and Explain why when it is subject to the initial condition u(x,0) = u0(x) is not well-posed.

Answer 1

ux= du(x,y)/dx

Answer 2

no it is not that easy! we need to use the method of characteristic

Answer 3

dx/ds=1 so x=s+c
and we usually use the initial condition x(0)=x' so c=x'

Answer 4

yes

Answer 5

i stuck because i don't know how to find s

Answer 6

dy/ds=0 so y=0 and by the initial condition y(0)=0 so c=0 so y=0

Answer 7

i need to find u(x,y)

Answer 8

u (x,0) = u_0 (x')= u_0 (x - s)
what is s here???

Answer 9

are you doing pde's by separation of variables?

is that the point of this?

Answer 10

yes

Answer 11

so \(u_x = Y X_x = 0 \implies X_x = 0, \implies X(x) = const\)

disregarding the trivial solution \(Y = 0\)

this next \(u(x,0) = u_o(x) \) bit is not clear in its meaning.