Maths. Section ADifferential Equations

Solution of differential equation x dy-y dx=0 represents:

Solution of differential equation $ x\,dy-y\,dx=0 $ represents:

Answer & explanation

Correct answer: option 1

The correct answer is Option (1) → Family of straight lines passing through origin

The variables separate:

$ \displaystyle x\,dy=y\,dx $

$ \displaystyle \frac{dy}{y}=\frac{dx}{x} $

Integrating,

$ \displaystyle \log|y|=\log|x|+\log|c| $

$ \displaystyle y=cx $

For every value of c, $ y=cx $ is a straight line through the origin with slope c.

Shortcut: Rewrite as $ \frac{dy}{dx}=\frac{y}{x} $: the slope at every point equals the slope of the line joining that point to the origin, which is true only of lines through the origin.

NCERT Exemplar:
Differential Equations

$NCERT extract$