If $y\left(x\right)$   is the solution of differential equation  $(x{y}^2+x)dx+(y-x^{2}y)dy=0$  and  $y\left(0\right)=0$ then its solution is 

$$\begin{gathered} x(y^2+1)dx+y(1-x^2)dy=0 \Rightarrow \frac{x}{1-x^2}dx+\frac{y}{1+y^2}dy=0 \\ \Rightarrow \int \frac{x}{1-x^2}dx+\int \frac{y}{1+y^2}dy=c \Rightarrow \frac{-1}{2}\log |1-x^2|+\frac{1}{2}\log |1+y^2|=\log c \\ \Rightarrow \frac{1}{2}\log \left|\frac{1+y^2}{1-x^2}\right|=\log c \Rightarrow \log \left|\frac{1+y^2}{1-x^2}\right|=\log c \Rightarrow 1+y^2=c(1-x^2) \\ y\left(0\right)=0\Rightarrow c=1 \Rightarrow x^2+y^2=0 \end{gathered}$$