Explanation:
(A) : The given family of curve,
\(y=a(x+a)^{2}\)
On differentiating w.r.t. \(x\), we get-
\(\frac{d y}{d x}=a \times 2(x+a) \Rightarrow \frac{d y}{d x}=2 a x+2 a^{2}\)
Again, differentiating w.r.t. \(x\), we get -
\(\frac{d^{2} y}{d x^{2}}=2 a+0 \Rightarrow a=\frac{1}{2} \frac{d^{2} y}{d x^{2}}\)
On putting the value of a in equation(i), we get-
\(\mathrm{y}=\frac{1}{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\left(\mathrm{x}+\frac{1}{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx})^{2}}\right.\)
\(y=\frac{1}{2} \frac{d^{2} y}{d x^{2}}\left[x^{2}+\frac{1}{4}\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+x \frac{d^{2} y}{d x^{2}}\right]\)
\(y=\frac{x^{2}}{2} \frac{d^{2} y}{d x^{2}}+\frac{1}{8}\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\frac{x}{2}\left(\frac{d^{2} y}{d x^{2}}\right)^{2}\)
So,
Order \(=2\)
Degree \(=3\)