Question

y ln x ln y dx + dy = 0

Answer 1

$$y\ln y\ln x dx+dy=0$$

$$\dfrac{dy}{y\ln y}=-\ln xdx$$

Integrate

$$\int\dfrac{dy}{y\ln y}=-\int\ln xdx$$

$$\int\ln xdx=x\ln x-\int x(\dfrac{1}{x})dx=x\ln x-x-\ln C$$

$$\ln(|\ln y|)=-x\ln x+x+\ln C$$

$$\ln y=C(e^{x-x\ln x})$$

$$y=e^{C(e^{x-x\ln x})}$$