Question

Solve the differential equation
xdx+ydy= a^2(xdy-ydx)/x^2+ y^2

Answer 1

$\displaystyle xdx+ydy = \frac{a^2(xdy-ydx)}{x^2+ y^2}\\ \mathrm{d}(x^2 +y^2) = \frac{a^2\frac{(xdy-ydx)}{x^2}}{1 + \frac{y^2}{x^2}}\\ \mathrm{d}(x^2 +y^2) = \frac{a^2\mathrm{d}\left(\frac{y}{x}\right)}{1 + \frac{y^2}{x^2}}\\ \int \mathrm{d}(x^2 +y^2) = \int \frac{a^2\mathrm{d}\left(\frac{y}{x}\right)}{1 + \left(\frac{y}{x}\right)^2}\\ x^2 +y^2 = a^2 \arctan\left(\frac{y}{x}\right) + C\\ \therefore x^2 +y^2 - a^2 \arctan\left(\frac{y}{x}\right) = C\\$