Hi Folks,

I am new to the method of characteristics for solving 1st order diff eqns. I am looking at an example in the book, which i do not quite follow..here goes

$\displaystyle x^2u_x+yu_y+xyu=1$

which is of the form $\displaystyle a(x,y)u_x+b(x,y)u_y+c(xy)u=f(x,y)$

The characteristic equation is $\displaystyle \frac{dy}{dx}=\frac{b}{a}=\frac{y}{x^2}$

which becomes $\displaystyle ln(y) + 1/x = k = \eta = \varphi(x,y)$

The bit I dont understand is making the transformation with

$\displaystyle \xi = x\text{and}\eta = ln(y) + 1/x$ to the partial differential eqn as above with $\displaystyle \omega(\xi, \eta)=u(x,y)$

$\displaystyle u_x = \omega_\xi\xi+\omega_\eta\eta_x = \omega_\xi+\omega_\eta(\frac{-1}{x^2})$

How is this $\displaystyle u_x$ calculated?