Question

xe^y dy + x^2+1/y dx= 0

Answer 1

$$xe^y dy + \dfrac{x^2+1}{y} dx= 0$$

$$ye^ydy=-\dfrac{x^2+1}{x}dx$$

Integrate both sides

$$\int ye^ydy=-\int\dfrac{x^2+1}{x}dx$$

$$\int ye^ydy=ye^y-\int e^ydy=ye^y-e^y+C_1$$

$$ye^y-e^y=-\dfrac{x^2}{2}-\ln x+C$$