Find the product of the order and the degree of the differential equation $\displaystyle \left[\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{xy}^{2}\right)\right] \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0$.

Given differential equation can be written as \[2 x y\left(\frac{d y}{d x}\right)^{2}+y^{2} \frac{d y}{d x}+y=0 \quad \text { Order }=1, \text { Degree }=2 \] Order × degree \(\displaystyle =1 \times 2=2\) \[\begin{aligned} \int \frac{\sin 3 x}{\sin x} d x & =\int \frac{3 \sin x-4 \sin ^{3} x}{\sin x} d x \\ & =\int\left[3-4 \frac{(1-\cos 2 x)}{2}\right] d x \\ & =\int(1+2 \cos 2 x) d x \\ & =x+\sin 2 x+C \end{aligned} \] Let \(\displaystyle e^{x}=t, e^{x} d x=d t\)