Question

Solve this ordinary differential equation dy/dx = (x^2-x+)/(y-1)

Answer 1

This equation is easily separable as

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{x^2-x+k}{y-1}\implies(y-1)\,\mathrm dy=(x^2-x+k)\,\mathrm dx$

where the $k$ term is assumed to be some constant that was omitted from the question. Integrating both sides gives

$\dfrac{y^2}2-y=\dfrac{x^3}3-\dfrac{x^2}2+kx+C$