The curve satisfying the differential equation ydx-(x+3y²)dy=0 and passing through the point (1,1) also passes through the point __________

Given: \(y\,dx-(x+3y^{2})\,dy=0 \;\Rightarrow\; y\,\dfrac{dx}{dy}=x+3y^{2}\).

So \( \dfrac{dx}{dy}-\dfrac{1}{y}x=3y\) (linear). Integrating factor \(=\exp\!\int\!-\dfrac{1}{y}dy=\dfrac{1}{y}\).

\(\displaystyle \frac{d}{dy}\!\left(\frac{x}{y}\right)=3 \;\Rightarrow\; \frac{x}{y}=3y+C \;\Rightarrow\; x=3y^{2}+Cy.\)

Through \((1,1)\): \(1=3(1)+C(1)\Rightarrow C=-2\). Hence curve: \(x=3y^{2}-2y\).