The general solution of the differential equation $(x^2-yx^2)dy + (y^2+x^2y^2)dx = 0$ is:

$\log_e|y|+\frac{1}{x}+\frac{1}{y}-x= c$, where c is constant of integration

The correct answer is Option (1) → $\log_e|y|+\frac{1}{x}+\frac{1}{y}-x= c$, where c is constant of integration

Given differential equation: $(x^2 - yx^2) \, dy + (y^2 + x^2 y^2) \, dx = 0$

Factor terms:

$(x^2(1 - y)) \, dy + y^2(1 + x^2) \, dx = 0$

Rewrite in separable form:

$\frac{1 - y}{y^2} \, dy + \frac{1 + x^2}{x^2} \, dx = 0$

Integrate each term:

$\int \left(\frac{1}{y^2} - \frac{1}{y}\right) dy + \int \left(\frac{1}{x^2} + 1\right) dx = 0$

$\int y^{-2} dy - \int y^{-1} dy + \int x^{-2} dx + \int dx = 0$

$-y^{-1} - \ln|y| - x^{-1} + x = C$

General solution: $x - \frac{1}{x} - \ln|y| - \frac{1}{y} = C$