This is Exercise 2.6.44 in Fundamentals of Differential Equations by Nagle.
Given an equation with linear coeff's
$$(a_1x+b_1y+c_1)dx+(a_2x+b_2y+c_2)dy=0$$
The problem is to show, when $a_1b_2=a_2b_1$, that it can be written in the form
$$\frac{dy}{dx}=G(ax+by)$$
When $a_1,a_2 \neq 0$, I can see how to do it:
$$\frac{dy}{dx}=- \frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}=- \frac{a_2}{a_1}\cdot \frac{a_1x+b_1y+c_1}{a_1x+b_1y+c_2a_1/a_2}$$
which is $G(a_1x+b_1y)$ for some $G$.
But what about the case $a_1=b_2=0$? Then it would look like
$$\frac{dy}{dx}=- \frac{b_1y+c_1}{a_2x+c_2}$$
and I don't know how to get it in the desired form. Nagle didn't say assume things weren't 0, so I assume he wanted us to consider this edge case? (But I can see that it is separable, so it is in a form I know how to solve).
Thanks a lot in advance