$$a \frac{\partial w}{\partial x}+b\frac{\partial w}{\partial y}+f(x,y)w+g(x,y)=0$$ thanks for any comment or possibly reference.
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2 Answers
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It is called the method of characteristics, have a look at http://www.stanford.edu/class/math220a/handouts/firstorder.pdf
Equation 2.19 on page 17 is a much more general formula to what you have. Set $a(x,y,u) = a$ (constant), same with $b(x,y,u) = b$ and $c(x,y,u) = -g(x,y) - f(x,y)u$ for your equation.
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You can check it out here for more details of partial differentiation http://www.math.hmc.edu/calculus/tutorials/partialdifferentiation/
