Method of characteristics for a 1st order quasi-linear PDE.

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Hi,

I'm looking over the examples in my book for this problem and the general approach is

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

where u(x,y)

I have the following problem in my notes:

1/x * u_x + 1/y * u_y = x^2 * sqrt(z)

and I get the solution easily because of the format:

Another problem i did was

y*u_x - x * u_y = 2xyu


where i multiplied both sides by 1/xy to get:

1/x*u_x - 1/y * u_y = 2u


the solution followed easily because of the classic format:

however, what i don't get is this format

u*u_x + y*u_y = x

the problem is an ivp with the following characteristics:

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

z = u(x_o,y_o) = u(s,s) = 2s

how do i get the solutin when it's not in the standard format? Or am i misinterpreting

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

in the sense that the coefficient functions (a(x,y,z), b(x,y,z), c(x.y,z) can be any variable x,y,u(x_o,y_o)?

I don't have the book and all my examples are written in the convenient form i'm after which confuses me

thank you
 

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