# Quasilinear pde

1. Homework Statement

Solve the following IVP for 1st order quasilinear PDE
s using the method of characteristics.

u*u_x + y*u_y = x

u = 2s, y = s, x = s

2. Homework Equations

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

z = u(x_o,y_o) = 2s

3. The Attempt at a Solution

The problem confuses me in the sense that this is the first one where x and y aren't coupled with their respective partial derivatives. for example I can solve

x*u_x + y*u_y = x*u

I'm just looking for more insight into the method of characteristics. does it imply that I can replace u with x with the parameters?

back to the original problem

u*u_x + y*u_y = x

u = 2s, y = s, x = s

is this an acceptable approach

dx/dk = u

and using the characteristic to write

dx/dk = 2x (where s is the first parameter, and k the second one because the equation has 2 independent vars)

dy/dk = y

dz/dk = x

this is the only part that confuses me and makes me believe im missing the bigger picture for this method.

Thanks!