Question

The general solution of the differential equation $\left( x \sin \frac{y}{x} \right) \frac{dy}{dx} = y \sin \frac{y}{x} - x$ is
Identify the correct option from the following:

• A. $\log x + \tan \frac{y}{x} = c$
• B. $\log x + \cos \frac{y}{x} = c$
• C. $\log x - \sin \frac{y}{x} = c$
• D. $\log x - \cos \frac{y}{x} = c$

Hint:

For homogeneous differential equations, use the substitution $v = \frac{y}{x}$ to reduce the equation to a separable form.

Answer 1

The Correct Option is D

Step 1: Recognize the form of the equation
The equation $\left( x \sin \frac{y}{x} \right) \frac{dy}{dx} = y \sin \frac{y}{x} - x$ is homogeneous. Let $v = \frac{y}{x}$, so $y = vx$, $\frac{dy}{dx} = v + x \frac{dv}{dx}$. Substitute: $\left( x \sin v \right) \left( v + x \frac{dv}{dx} \right) = (vx) \sin v - x$, simplify: $x \sin v \cdot x \frac{dv}{dx} = -x$, $\sin v \frac{dv}{dx} = -\frac{1}{x}$. Step 2: Solve the separated equation
$\sin v \, dv = -\frac{dx}{x}$, integrate: $-\cos v = -\log x + c$, $\log x - \cos v = c$. Substitute back: $\log x - \cos \frac{y}{x} = c$. Step 3: Match with options
The solution $\log x - \cos \frac{y}{x} = c$ matches option (4).