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. $\frac{d^{2}v}{d{r}^2}-\frac{2}{r}\frac{dv}{dr}=0$
• C. $\frac{d^{2}v}{d{r}^2}+\frac{2}{r}\frac{dv}{dr}=0$
• D. None of these

Answer 1

The correct option is C

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

$\frac{dv}{dr}=-\frac{A}{r^2}+0\Rightarrow \frac{d^2}{d{r}^2}=\frac{2A}{r^3}\Rightarrow \frac{d^{2}v}{d{r}^2}=\frac{2}{r}\left(\frac{A}{r^2}\right)\Rightarrow \frac{d^{2}v}{d{r}^2}=\frac{2}{r}(-\frac{dv}{dr})\Rightarrow \frac{d^{2}v}{d{r}^2}+\frac{2}{r}\frac{dv}{dr}=0.$