Integrating factor ( I. F ) of $ydx-xdy+(x^2+y^2)dx=0$.

Question

Find the integrating factor of the equation: $y\,\mathrm dx - x\,\mathrm dy + (x^2 + y^2)\,\mathrm dx = 0$

Answer 1

Rather than calculating the integrating factor we should go for direct solution. Anyway we need I.F. for solving the Exact Differential Equation. So here is it's solution;

$$ydx-xdy+(x^2+y^2)dx=0$$ $$\frac {ydx-xdy}{(x^2+y^2)}+dx=0$$ $$d(\arctan(\frac xy))+dx=0$$ $$\arctan(\frac xy)+x=C$$