How Do Auxiliary Equations Simplify Complex Mathematical Problems?

[parkcman]

I'm flipping through a math book to eventually be able to get to a Fourier numerical solutions but the second chapter has me puzzled
Ux = partial d of U wrt x
Uy = partial d o U wrt y

a*Ux + b*Uy + k*U(x,y) = f(x,y)
dU = Ux * dx + Uy * dy

then they get
dx/a = dy/b = du/ (f(x,y) - k*U)

i don't seem to understand how there is an equal sign in between dx/a and dy/b or where it arises from
when i try to go through the math i seem to get a + sign but the math keeps continuing along with it and they call it "axuiliary equation" and google doesn't seem to help

Thanks for reading

 

[Greg Bernhardt]

Auxiliary equations can be used to solve simple equations, as well as very complicated equations. The auxiliary equation used to solve an equation is called an "inverse equation" to the equation.

An auxiliary equation is an equation that is used to solve a more complicated equation.