The solution of the differential equation $\mathrm{ydx}-\mathrm{xdy}+{\mathrm{xy}}^2 \mathrm{dx}=0,$ is

Given equation is, $\mathrm{ydx}-\mathrm{xdy}+{\mathrm{xy}}^2 \mathrm{dx}=0$

Which should be converted into exact form

$$\begin{gathered} \mathrm{i}.\mathrm{e}. \frac{\mathrm{ydx}-\mathrm{xdy}}{{\mathrm{y}}^2}+\mathrm{xdx}=0 \\ \Rightarrow \mathrm{d}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)+\mathrm{d}\left(\frac{{\mathrm{x}}^2}{2}\right)=0 \end{gathered}$$

Integrating both the sides, we get

$$\begin{gathered} \frac{\mathrm{x}}{\mathrm{y}}+\frac{{\mathrm{x}}^2}{2}=\mathrm{constant} \\ \mathrm{or} \frac{\mathrm{x}}{\mathrm{y}}+\frac{{\mathrm{x}}^2}{2}=\mathrm{\lambda } \end{gathered}$$

Hence, (b) is the correct answer.