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Math Help - New co ordinates to develope 1st order characteristic equations

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    Senior Member bugatti79's Avatar
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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

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

     w(s,t)=u(x(s,t),y(s,t)).

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

    I think these are the characeristic equations...

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

    Attached is my attempt but its not fruitful...
    Attached Thumbnails Attached Thumbnails New co ordinates to develope 1st order characteristic equations-imag0182.jpg  
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