Question

The general solution of the differential equation \( \left(x \sin \frac{y}{x} \right) dy = \left( y \sin \frac{y}{x} - x \right) dx \) is

• A. \( \cos \left( \frac{y}{x} \right) = \log |x| + c \)
• B. \( \cos \left( \frac{y}{x} \right) = \frac{1}{x} + c \)
• C. \( \cos \left( \frac{x}{y} \right) = \log |y| + c \)
• D. \( \cos \left( \frac{y}{x} \right) = \frac{2}{x} + c \)

Hint:

In differential equations involving terms like \( \frac{y}{x} \), try the substitution \( v = \frac{y}{x} \). This often converts the equation into a separable form.

Answer 1

The Correct Option is A

We are given: \[ \left(x \sin \frac{y}{x} \right) dy = \left( y \sin \frac{y}{x} - x \right) dx \] Step 1: Rewrite the equation. \[ x \sin \left( \frac{y}{x} \right) \, dy = \left[ y \sin \left( \frac{y}{x} \right) - x \right] dx \] Step 2: Use the substitution \( v = \frac{y}{x} \Rightarrow y = vx \Rightarrow dy = v\,dx + x\,dv \) Substitute in the equation: \[ x \sin(v) (v\,dx + x\,dv) = \left( vx \sin(v) - x \right) dx \] Expand both sides: \[ xv \sin(v) \, dx + x^2 \sin(v) \, dv = x v \sin(v) \, dx - x \, dx \] Cancel \( x v \sin(v) dx \) from both sides: \[ x^2 \sin(v) \, dv = -x \, dx \] Step 3: Separate variables. \[ x \sin(v) \, dv = - \, dx \] Step 4: Integrate both sides. \[ \int \sin(v) \, dv = -\int \frac{1}{x} \, dx \] \[ - \cos(v) = - \log |x| + c \Rightarrow \cos\left( \frac{y}{x} \right) = \log |x| + c \] % Final Answer \[ \boxed{ \cos\left( \frac{y}{x} \right) = \log |x| + c } \]