Question

Solution of Differential Equation $xdy - ydx = 0$ represents

• A. A rectangular Hyperbola
• B. Parabola whose vertex is at origin
• C. Straight line passing through origin
• D. A circle whose centre is origin

Answer 1

The Correct Option is C

To solve the differential equation \(x \, dy - y \, dx = 0\), let's follow the steps:

Rearrange the terms to facilitate separation of variables: 

\(x \, dy = y \, dx\)

Divide both sides by \(xy\) to separate the variables:

\(\frac{dy}{y} = \frac{dx}{x}\)

Integrate both sides:

\(\int \frac{dy}{y} = \int \frac{dx}{x}\)

The integrals yield:

\(\ln|y| = \ln|x| + C\)

where \(C\) is the constant of integration.

Exponentiate both sides to eliminate the logarithm:

\(e^{\ln|y|} = e^{\ln|x| + C}\)

which gives:

\(|y| = e^{C}|x|\)

Let \(e^{C} = k\) (where \(k\) is a positive constant), then:

\(|y| = k|x|\)

This implies:

\(y = kx\) or \(y = -kx\), representing straight lines through the origin with slopes \(k\) and \(-k\) respectively.

Thus, the solution of the differential equation \(x \, dy - y \, dx = 0\) indeed represents a straight line passing through the origin.

This justifies the correct answer as: Straight line passing through the origin.