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=

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

Tardigrade Admin · Accepted Answer

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} \dots

.....(i)

∴ 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} (\frac{d y}{d x})^{2}

∴

order

= m = 1

and degree

= n = 2

∴ mn - m + n = 1 \times 2 - 1 + 2 = 3.

Q. 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 mn−m+n=

1

4

3

2

The equation of the family of parabolas with focus at the origin and X -axis as its axis is given by y2=4a(x+a)=4ax+4a2… .....(i) ∴2ydxdy​=4a ⇒a=21​ydxay​.....(ii) From Eqs. (i) and (ii), we have y2=2xydxdy​+y2(dxdy​)2 ∴ order =m=1 and degree =n=2 ∴mn−m+n=1×2−1+2=3.

Q. 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 $mn - m + n =$