I came across this problem today in Differential Equations with Applications and Historical Notes by George F. Simons, and I almost do not know how to start off. I need help. It says,
Show that the substitution z = ax + by + c changes y′ = f (ax + by + c) into an equation with separable variables.
Just substitute $z=ax+by+c \to z'=by'+a$
$$y′ = f (ax + by + c)$$ $$y' =f(z)$$ For $b \ne 0$ $$\frac {z'-a}{b}=f(z)$$ $${z'}=bf(z)+a$$ this form of the equation is separable $$\int\frac {dz}{bf(z)+a}=\int dx=x+K$$
Edit for $b=0$ It's already separable.. $$y′ = f (ax + c)$$ $$\int dy =\int f (ax + c)dx $$ $$y =\int f (ax + c)dx $$