The differential equation y d y d x + x = C represents:

Question

The differential equation $y \frac{d y}{d x} + x = C$ represents:

Smart Achievers · Accepted Answer

(4)

Ans. family of circles

Explanation :

Given that,

$y \frac{d y}{d x} + x = C$

$\Rightarrow$ $y \frac{d y}{d x} = C - x$

$\Rightarrow$ $y d y = (C - x) d x$

On integrating both sides, we get

$\frac{y^{2}}{2} = C x - \frac{x^{2}}{2} + K$ (where $K$ is an integration constant)

$\Rightarrow$ $\frac{x^{2}}{2} + \frac{y^{2}}{2} = C x + K$

$\Rightarrow$ $\frac{x^{2}}{2} + \frac{y^{2}}{2} - C x = K$

which represents a family of circles.

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