Some DEs become separable with the right substitution. Type 1: dy/dx = f(ax + by + c). Substitute v = ax + by + c, then dv/dx = a + bdy/dx = a + bf(v). This is separable in v and x. Type 2: dy/dx = f$\frac{(a1*x + b1*y + c1}{(a2*x + b2*y + c2)}$) where a1/a2 = $\frac{b1}{b2}$. Substitute the linear combination as a new variable. Type 3: dy/dx = f$\frac{(a1*x + b1*y + c1}{(a2*x + b2*y + c2)}$) where a1/a2 != $\frac{b1}{b2}$. Shift origin to the intersection point of a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 to eliminate c1, c2, then the equation becomes homogeneous.