Question

The differential equation of the family of curves $v=\frac{A}{r}+B$, where $A$ and $B$ are arbitrary constants is

• A. $\frac{d^{2}v}{d{r}^2}+\frac{1}{r}\frac{dv}{dr}=0$
• B. $\frac{d^{2}v}{d{r}^2}-\frac{2}{v}\frac{dv}{dr}=0$
• C. $\frac{d^{2}v}{d{r}^2}+\frac{2dv}{rdr}=0$
• D. $\frac{d^{2}v}{d{r}^2}+\frac{2}{v}\frac{dv}{dr}=0$

Answer 1

The correct option is C $\frac{d^{2}v}{d{r}^2}+\frac{2dv}{rdr}=0$

$V=\frac{A}{r}+B$

On differentiating ,we get

$\frac{dv}{dr}=\frac{-A}{r^2}+0$=$\frac{-A}{r^2}$

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

On differentiating, we get

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

That is,

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