Question

Solve :$(x^2-y{x}^2)dy+(y^2+x^2{y}^2)dx=0$

Answer 1

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

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

$=x^2\left(1-y\right)dy=-y^2\left(1+x^2\right)dx$

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

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

$=\frac{-1}{y}-\log y=-\left(\frac{-1}{x}+x\right)+c$

$=\frac{-1}{y}-\log y=\frac{1}{x}-x+c$

$=\frac{1}{x}+\frac{1}{y}-x+\log y+c$