Solution of the differential equation $x d y - y d x = 0$ represents.

A parabola whose vertex is at origin.

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A circle whose centre is at origin.

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A rectangular hyperbola.

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A straight line passing thorugh origin.

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Solution

The correct option is C A straight line passing thorugh origin.
Given, $x d y - y d x = 0$
$\Rightarrow \frac{d y}{y} - \frac{d x}{x} = 0$
On integrating, we get
$l o g y - l o g x = l o g c$
$\Rightarrow l o g \frac{y}{x} = l o g c$
$\Rightarrow \frac{y}{x} = c$
$\Rightarrow y = c x$ , which is a straight line passing through the origin.

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Q. What is the solution of the differential equation

x d y + y d x = 0

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Solution of the differential equation xdy−ydx=0 represents.

The correct option is C A straight line passing thorugh origin.Given, xdy−ydx=0⇒dyy−dxx=0On integrating, we get logy−logx=logc⇒logyx=logc⇒yx=c⇒y=cx, which is a straight line passing through the origin.

Solution of the differential equation $x d y - y d x = 0$ represents.