Question

Solve the homogeneous differential equation 𝑥𝑦 𝑑𝑦/𝑑𝑥 = 𝑦² + 𝑥² 𝑑y/𝑑𝑥   .

             

Answer 1

$xy\frac{dy}{dx}=y^2+x^2\frac{dy}{dx}\\y=zx\\y'=z'x+z\\xzx\left( z'x+z \right) =z^2x^2+x^2\left( z'x+z \right) \\zz'x+z^2=z^2+z'x+z\\z'\left( z-1 \right) x=z\\\frac{z-1}{z}dz=\frac{dx}{x}\\z-\ln \left| z \right|+C'=\ln \left| x \right|\\\frac{y}{x}-\ln \left| y \right|+\ln \left| x \right|+C'=\ln \left| x \right|\\\frac{y}{x}-\ln \left| y \right|=Const$