First Order ODE in form y' = f (a x + b y + c)

Theorem
The first order ODE:
 * $\dfrac {\d y} {\d x} = \map f {a x + b y + c}$

can be solved by substituting:
 * $z := a x + b y + c$

to obtain:
 * $\ds x = \int \frac {\d z} {b \map f z + a}$

Proof
We have:
 * $\dfrac {\d y} {\d x} = \map f {a x + b y + c}$

Put:
 * $z := a x + b y + c$

Then:

This can be solved by Separation of Variables:
 * $\ds x = \int \frac {\d z} {b \map f z + a}$