Find the differential equation representing the family of curves $v=\dfrac{A}{r}+B$, where $A$ and $B$ are arbitrary constants.

$r\dfrac{d^{2}v}{dr^{2}}+2\dfrac{dv}{dr}=0$

CBSE Class 12 · 2015 · Differential EquationsVery Short Answer

Find the differential equation representing the family of curves

v = \frac{A}{r} + B

, where

A

and

B

are arbitrary constants.

Answer: $r \frac{d ^{2} v}{d r ^{2}} + 2 \frac{d v}{d r} = 0$

Two arbitrary constants (

A, B

) need two differentiations.
Differentiate

v = \frac{A}{r} + B

w.r.t.

r

\frac{d v}{d r} = - \frac{A}{r ^{2}} \Rightarrow r^{2} \frac{d v}{d r} = - A

Differentiate again:

r^{2} \frac{d ^{2} v}{d r ^{2}} + 2 r \frac{d v}{d r} = 0

Divide by

r

r \frac{d ^{2} v}{d r ^{2}} + 2 \frac{d v}{d r} = 0

CBSE Class 12 Board · 2015 · Subjective