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New co ordinates to develope 1st order characteristic equations

Intorduce new variables s=s(x,y) and t=t(x,y) in the equation

$\displaystyle a(x,y)u_x+b(x,y)u_y=c(x,y,u)$ and derive the corresponding first order equation for w, where

$\displaystyle w(s,t)=u(x(s,t),y(s,t)).$

Show that requiring the coefficient of $\displaystyle w_t$ to be zero leads to the equations for characteristics

I think these are the characeristic equations...

$\displaystyle \displaystyle \frac{dx}{dt}=a(x,y), \frac{dy}{dt}=b(x,y).$

Attached is my attempt but its not fruitful...