Question

The differential equation $y\frac{dy}{dx}+x=a$ where '$a$' is any constant represents:

• A. A set of straight lines
• B. A set of ellipses
• C. A set of circles
• D. None of the above

Answer 1

Answer: (C)

A set of circles

$y\frac{dy}{dx}+x=a$

$\Rightarrow ydy+xdx=adx$

Integrating both sides,

$\Rightarrow \frac{y^2}{2}+\frac{x^2}{2}=ax+c\Rightarrow x^2+y^2-2ax-2c=0\Rightarrow {(x-a)}^2+y^2=a^2+2c$

Clearly, this equation represents circle with centre $(a,0)$ and radius $\sqrt{a^2+2c}.$
Option C is correct.