[tex] \displaystyle
\dfrac{dy}{dx}=\dfrac{xy^2+x}{y}\\
\dfrac{dy}{dx}=\dfrac{x(y^2+1)}{y}\\
\dfrac{y}{y^2+1}\, dy=x\, dx\\
\int \dfrac{y}{y^2+1}\, dy=\int x\, dx\\
\dfrac{\ln (y^2+1)}{2}=\dfrac{x^2}{2}+C\\
\ln(y^2+1)=x^2+C\\
y^2+1=e^{x^2+C}\\
y^2=e^{x^2+C}-1\\
y=\sqrt{e^{x^2+C}-1} \vee y=-\sqrt{e^{x^2+C}-1}\\
y=\sqrt{Ce^{x^2}-1} \vee y=-\sqrt{Ce^{x^2}-1}\\\\
0=\sqrt{Ce^{0^2}-1} \vee 0=-\sqrt{Ce^{0^2}-1}\\
0=\sqrt{C-1} \vee 0=-\sqrt{C-1}\\
C-1=0\\
C=1
[/tex]
[tex] \boxed{y=\sqrt{e^{x^2}-1} \vee y=-\sqrt{e^{x^2}-1}} [/tex]
