(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

u*u_x + y*u_y = x

Initial condition:

u = 2*s on the parametric curve given by x = s, y = s, s is any real number.

2. Relevant equations

Given the equation:

a(x,y,z)*u_x + b(x,y,z)*u_y = c(x,y,z)

Here, u(x,y) is an unknown function which we're trying to find.

a(x,y,z),b(x,y,z),c(x,y,z) are given functions of 3 variables.

The initial condition is a requirement that the unknown function u has a prescribed value when restricted to the parametric curve:

u(x_o(s),y_o(s)) = u_o(s), u_o(s) is a given function.

3. The attempt at a solution

My attempt crumbles early on.

From looking at the equation I get:

a(x,y,z) = u

b(x,y,z) = y

c(x,y,z) = x

We alos have x(0) = s, y(0) = s, and u(x(0),y(0)) = u(s,s) = 2s

My professor mentioned we need to introduce a 2nd variable, t, for the following relations

dx(t)/dt = a(x(t),y(t),z(t))

dy(t)/dt = b(x(t),y(t),z(t))

dz(t)/dt = c(x(t),y(t),z(t))

1.dx/dt = u , x(0) = s

2.dy/dt = y , y(0) = s

3.dz/dt = x , u(x(0),y(0)) = 2s

I have a feeling that this might be wrong, and my other hypothesis is the following set up:

dz(t)/dt = a(x(t),y(t),z(t))

dy(t)/dt = b(x(t),y(t),z(t))

dx(t)/dt = c(x(t),y(t),z(t))

*note the swap between z and x.

my main problem is relating a,b,c to the equation. For example, are

a(x(t),y(t),z(t))

b(x(t),y(t),z(t))

c(x(t),y(t),z(t))

fixed with respect to the partial derivattives u_x and u_y, or can I switch them around like I did in my second hypothesis?

This clarification will allow me to finish the problem. Any input is appreciated. Thank you!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: 1st order quasi-linear pde

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**