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

Theorem
The first order ODE:
 * $\dfrac {\mathrm d y} {\mathrm d x} = f \left({a x + b y + c}\right)$

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

to obtain:
 * $\displaystyle x = \int \frac {\mathrm d z} {b f \left({z}\right) + a}$

Proof
We have:
 * $\dfrac {\mathrm d y} {\mathrm d x} = f \left({a x + b y + c}\right)$

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

Then:

This can be solved by Separation of Variables:
 * $\displaystyle x = \int \frac {\mathrm d z} {b f \left({z}\right) + a}$