Question

Solution of Differential Equating 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

The given differential equation is: \[ xdy - ydx = 0 \] Rearranging terms: \[ \frac{dy}{dx} = \frac{y}{x} \] This is a separable differential equation. By separating variables: \[ \frac{dy}{y} = \frac{dx}{x} \] Integrating both sides: \[ \ln|y| = \ln|x| + C \] This implies: \[ y = Cx \]

So, the correct answer is (C) : Straight line passing through origin.

Answer 2

We are given the differential equation: 

\[ x\,dy - y\,dx = 0 \]

Step 1: Rearranging the equation

Divide both sides by \( dx \): \[ x \frac{dy}{dx} - y = 0 \Rightarrow \frac{dy}{dx} = \frac{y}{x} \]

Step 2: Solve the differential equation

This is a separable differential equation: \[ \frac{dy}{y} = \frac{dx}{x} \] Integrating both sides: \[ \int \frac{dy}{y} = \int \frac{dx}{x} \Rightarrow \ln |y| = \ln |x| + C \] Taking exponentials on both sides: \[ |y| = A |x| \Rightarrow y = Cx \quad \text{(where } C \text{ is a constant)} \]

Conclusion: The solution represents a family of straight lines passing through the origin.

Final Answer: Straight line passing through origin