Question

solve x^2du/dx+y^2du/dy=0

Answer 1

The subsidiary equations are given by

$$\dfrac{dx}{x^2}=\dfrac{dy}{y^2}=\dfrac{du}{0}$$

$$\dfrac{dx}{x^2}=\dfrac{dy}{y^2}$$

$$\int\dfrac{dx}{x^2}=\int\dfrac{dy}{y^2}$$

$$-\dfrac{1}{x}=-\dfrac{1}{y}+C_1$$

$$\dfrac{1}{y}-\dfrac{1}{x}=C_1$$

$$\dfrac{dx}{x^2}=\dfrac{du}{0}$$

$$du=0=>u=C_2$$

The desired solution is $f(C_1, C_2)=0$ or

$$f(\dfrac{1}{y}-\dfrac{1}{x}, u)=0$$