Question

The differential equation of the family of curves \[v = \frac{A}{r} + B\], where \[{\rm{A}}\]and \[{\rm{B}}\] are arbitrary constants, is
A. \[\frac{{{d^2}v}}{{d{r^2}}} + \frac{1}{r}\frac{{dv}}{{dr}} = 0\]
В. \[\frac{{{d^2}v}}{{d{r^2}}} - \frac{2}{r}\frac{{dv}}{{dr}} = 0\]
C. \[\frac{{{d^2}v}}{{d{r^2}}} + \frac{2}{r}\frac{{dv}}{{dr}} = 0\]
D. None of these

Answer 1

Hints:
With two variables A and B in the equation, the relationship between v and r is shown. As a result, in order to eliminate A and B and obtain the necessary differential equation, we must twice differentiate v with respect to r.
Formula used:
\[y = {x^n}\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = n{x^{n-1}}\]
Complete step-by-step solution:
We have been given the equation for the family of curves as below, where A, B are arbitrary constants.
Since,\[{\rm{v}} = \frac{{\rm{A}}}{{\rm{r}}} + {\rm{B}}\]-- (equation 1)
On differentiating both sides of equation (1) with respect to \[{\rm{r}}\], we get
\[\frac{{dv}}{{dr}} = \frac{{ - A}}{{{r^2}}} + 0 = \frac{{ - A}}{{{r^2}}}\]
Let's restructure the equation above to specifically get ‘A’ on one side.
The result is as follows:
\[A = - {r^2}\frac{{dv}}{{dr}}\]-- (equation 2)
Again, on differentiating the equation (2) on both sides with respect to \[{\rm{r}}\], we get
\[0 = - 2r\frac{{dv}}{{dr}} - {r^2}\frac{{{d^2}v}}{{d{r^2}}}\]
We have eliminated both the arbitrary constants A and B. 
The above equation can be rewrite as,
\[\frac{{{d^2}v}}{{d{r^2}}} + \frac{{2dv}}{{rdr}} = 0\]
Thus, the equation obtained is the differential equation representing the family of given curves.
Hence, the option C is correct.
Notes:
The student needs to remember that one must differentiate the equation as many times as there are arbitrary constants whenever one needs to get the differential equation of a family of curves. There cannot be any arbitrary constants in the differential equation. Students frequently make the error of removing only a small number of constants and creating a differential equation that contains one or more constants. This is entirely incorrect.