Question

If $m$ and $n$ are respectively the order and degree of the differential equation of the family of parabolas with focus at the origin and $X$ -axis as its axis, then $m n-m+n=$

• A. 1
• B. 4
• C. 3
• D. 2

Answer 1

Answer: (C)

The equation of the family of parabolas with focus at the origin and $X$ -axis as its axis is given by
$y^{2} =4 a(x+a)=4 a x+4 a^{2} \ldots$ .....(i)
$\therefore 2 \,y \frac{d y}{d x} =4 a$
$\Rightarrow a=\frac{1}{2} y \frac{a y}{d x}$.....(ii)
From Eqs. (i) and (ii), we have
$y^{2}=2 x y \frac{d y}{d x}+y^{2}\left(\frac{d y}{d x}\right)^{2}$
$\therefore $ order $=m= 1$ and degree $=n=2$
$\therefore m n-m+n=1 \times 2-1+2=3 .$