The solution of the equation \({\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{d
| The solution of the equation \({\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}} + {\rm{y}} = 0{\rm{}}\) passing through the point (1,1) is
A. x
B. x<sup>2</sup>
C. x<sup>-1</sup>
D. x<sup>-2</sup>
Please scroll down to see the correct answer and solution guide.
Right Answer is: C
SOLUTION
Given:
x dy/dx + y = 0, (x,y) = (1,1)
x dy/dx = -y
On rearranging,
dy/y = -dx/x
\(\smallint \frac{1}{y}dy = \smallint - \frac{1}{x}dx\)
On integrating,
lny = -lnx + c
Since equation passing through the point (1,1), therefore when x=1, y=1
0 = 0 + c
⇒ c = 0
\({\rm{lny}} = - {\rm{lnx}} = \frac{1}{{{\rm{lnx}}}}\) or y = 1/x
∴ The solution of the equation \({\bf{x}}\frac{{{\bf{dy}}}}{{{\bf{dx}}}} + {\bf{y}} = 0\) is x-1