solving a differential equation involving $\frac{y-x^2}{\sin y-x}$

Question

I'm trying to find the general solution to

$$\frac{\text{d}y}{\text{d}x} = \frac{y-x^2}{\sin y-x}$$

Any ideas would be greatly appreciated.

Thanks!

Answer 1

Your equation is exact once you write it as $$f(x,y)\,\mathrm d x+g(x,y)\,\mathrm d y=0.$$ Find a potential, and voilà. I'll leave you the fun of doing that; the general solution is implictly defined by the equation $$\frac{x^3}{3}-xy-\cos y=c$$ with $c$ a constant.