Explanation:
(A) : The given differential equation,
\(x=1+x y\left(\frac{d y}{d x}\right)+\frac{x^{2} y^{2}}{2}\left(\frac{d y}{d x}\right)^{2}+\frac{x^{3} y^{3}}{6}\left(\frac{d y}{d x}\right)^{3}+\ldots\)
Let, \(t=\left(x y \frac{d y}{d x}\right)\) \(\mathrm{x}=1+\frac{\mathrm{t}}{1 !}+\frac{\mathrm{t}^{2}}{2 !}+\frac{\mathrm{t}^{3}}{3 !}+\ldots . . \Rightarrow\left[\because \mathrm{e}^{\mathrm{x}}=1+\frac{\mathrm{x}}{1 !}+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{3}}{3 !}+\ldots \infty\right]\) So,
\(\mathrm{x}=\mathrm{e}^{\mathrm{t}} \tag{i}\)
Taking log both sides, we get-
\(\log x=\log _{e} e^{t}\)
\(\log x=t \log _{e} e\)
\(\log x=t\)
On putting the value of \(t\), we get-
\(\log x=x y \frac{d y}{d x}\)
Hence, Order \(=1\)
\(\text { Degree }=1\)