How to obtain the last ratio $\frac{d(x+z)}{(x+z)}$

Question

I am studying example-2.3

In the first line, it says that "the numerators and denominators in the first and last ratio"

And the following is obtained $$\frac{d(x+z)}{x+z}=\frac{dy}{y}$$

But I dont understand this statement. How to obtain the last ratio $\frac{d(x+z)}{(x+z)}$ Please, explain this more clear. Thank you :)

Answer 1

The writer has literally done as stated and "added the numerators and denominators" from the terms $\frac {dx}{y+z}$ and $\frac {dz}{x-y}$:

$${dx+dz\over y+z + x-y}={d(x+z)\over x+z}$$

Ordinarily, this would be a significant error in addition of fractions, but the author is making a point using this particular ratio in differential equation terms.

Putting the results together in this way creates extra meaning and equations to use in solving the original system of differential equations.