Question

Solve the differential equation

(x² + y²)dx-xydy = 0

Answer 1

Solution

Given equation may be rewritten in the form

$\frac{dy}{dx}=\frac{x^2+y^2}{xy}=\frac{x}{y}+\frac{y}{x}$

Let z=y/x => $\frac{dy}{dx}=\frac{dz}{dx}x+z$  => $\frac{dz}{dx}x+z\ =\frac{1}{z}+z$  => zdz = dx/x  =>   $\int{zdz=\int\frac{dx}{x}}$    => z²/2=ln|x|+C   =>  C = z²/2 - ln|x| = y²/(2x²)-ln|x|   =>  $y(x)=\sqrt{2x^2\left(C+ln|x|\right)}$

Answer

$y(x)=\sqrt{2x^2\left(C+ln|x|\right)}$