Question

The general solution of the differential equation. \[ \left( \frac{y}{x} \right) \cos \left( \frac{y}{x} \right) dx - \left[ \left( \frac{x}{y} \right) \sin \left( \frac{y}{x} \right) + \cos \left( \frac{y}{x} \right) \right] dy = 0 \] is

• A. \( y^2 \sin \left( \frac{y}{x} \right) = k \)
• B. \( x \sin \left( \frac{y}{x} \right) = k \)
• C. \( \sin \left( \frac{y}{x} \right) = k \)
• D. \( y \sin \left( \frac{y}{x} \right) = k \)

Hint:

For homogeneous differential equations of the form \(f(y/x)dx + g(y/x)dy = 0\), the substitution \(y = vx\) always works. After substitution, carefully collect coefficients of \(dx\) and \(dv\) before separating variables.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The given differential equation is homogeneous in \(y/x\). Substitute \(y = vx\) to reduce it to a separable form.

Step 2: Key Formula or Approach:
Let \(v = \frac{y}{x}\), so \(y = vx\) and \(dy = v\,dx + x\,dv\). Substitute into the equation and separate variables.

Step 3: Detailed Explanation:
Given: \[ v \cos v \, dx - \left( \frac{1}{v}\sin v + \cos v \right) dy = 0. \] Substitute \(dy = v\,dx + x\,dv\): \[ v \cos v \, dx - \left( \frac{1}{v}\sin v + \cos v \right)(v\,dx + x\,dv) = 0. \] Expand: \[ v \cos v \, dx - \frac{1}{v}\sin v \cdot v\,dx - \frac{1}{v}\sin v \cdot x\,dv - \cos v \cdot v\,dx - \cos v \cdot x\,dv = 0. \] Simplify: \[ v \cos v \, dx - \sin v \, dx - \frac{x}{v}\sin v \, dv - v \cos v \, dx - x \cos v \, dv = 0. \] The terms \(v \cos v \, dx\) and \(-v \cos v \, dx\) cancel. Thus: \[ - \sin v \, dx - x \left( \frac{\sin v}{v} + \cos v \right) dv = 0. \] Multiply by \(-1\): \[ \sin v \, dx + x \left( \frac{\sin v}{v} + \cos v \right) dv = 0. \] Divide by \(x \sin v\): \[ \frac{dx}{x} + \left( \frac{1}{v} + \cot v \right) dv = 0. \] Integrate: \[ \int \frac{dx}{x} + \int \frac{1}{v} dv + \int \cot v \, dv = \ln k. \] \[ \ln x + \ln v + \ln(\sin v) = \ln k. \] \[ \ln(x v \sin v) = \ln k \implies x v \sin v = k. \] Substitute back \(v = \frac{y}{x}\): \[ x \cdot \frac{y}{x} \cdot \sin\left( \frac{y}{x} \right) = k \implies y \sin\left( \frac{y}{x} \right) = k. \]

Step 4: Final Answer:
Option (D) is correct.