Homework Help: PDE question

Hey,

I'm trying to solve the following pde,

u(x,y) u_x + u_y =0 with u(x,0) = p(x) for some known p(x)

where u_x defines the partial derivative of u(x,y) wrt x

after finding the characteristic curves and the first integrals i get the general solution is

F(x^2 - zy^2, z) = 0

(note z=u(x,y))

At this point i'm not sure what to do next,

usually you can rewrite this as

f(x^2-zy^2) + z =0, however as z is contained in the argument you cant solve this for z without knowing what f is.

One thing i was thinking was just to make up some F and continue on, but that does not seem correct,

can anyone push me in the right direction?

 

dx/dt = u = z => x=zt + a

t=(x-a)/c

dy/dt = 1 => y=t+b

t=y-b

dz/dt = 0 => z=c


then dx/dt - z dy/dt = 0 => 1/2 d/dt(x^2 + zy^2) = 0

so x^2 + zy^2 = k


unless i made a dumb error, i thought it was pretty straight forward so left it out

 

oh ok,

the way they taught our class was what i posted above

dx/dt = u = z

dy/dt = 1

dz/dt = 0 => z=c


then dx/dt - z dy/dt = 0 => 1/2 d/dt(x^2 + zy^2) = 0

since dx/dt - z dy/dt= 1/2 d/dt(x^2 + zy^2)

so d/dt(x^2 + zy^2) = 0

integrating both sides gives x^2 + zy^2 = k

but yea there's lots of possibly first integrals,

the problem im having is the next step,

so using what you got

the general solution would be

F(x-uy, u) = 0

how can i solve this for u?

this is like

h(x-uy) + g(u) = 0 or something right?

im having trouble working out how to seperate u from everything else without having to define F

is it possible?

 

Oh,

Sorry I didn't mention it states in the question that p(x) is a smooth function,

Does this mean that you apply the initial condition to F(....)=0?

So for x=x y=0 you get


F(x,p(x)) = 0

f(x) + p(x) = 0

f(x) = -p(x)

f(x-uy) = -p(x-uy)

or something like that? i still see myself getting stuck alone the line

 

how come you write it as F(x-uy) and not F(x-uy,u) like i did below?

F(x-uy, u)=0

f(x-uy) = -u

at y=0

f(x)=-p(x)

i guess you can absorb the negative into p(x)

f(x-uy) = p(x-uy)

i guess one important question that im not sure about is,

what is the significance of saying that p(x) is smooth?

 

Ah right,

thanks for your help voko,

hey i think this pde is the inviscid burgers equation

http://en.wikipedia.org/wiki/Burgers'_equation#Inviscid_Burgers.27_equation