Question

Find the differential equation representing the family of curves $\nu =\frac{A}{r}+B$, where $A$ and $B$ are arbitary constants.

Answer 1

Given family of curves is $v=\frac{A}{r}+B$.

Differentiating once, we get:

$\frac{dv}{dr}=-\frac{A}{r^2}$

$\Rightarrow r^2\frac{dv}{dr}=-A$

Differentiating once more, we get:

$\frac{d^{2}v}{d{r}^2}=\frac{2A}{r^3}$

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

This is the required differential equation representing the given family of curves.