Question

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

• A. \(\cos \frac{x}{y} = \log_ex + c\)
• B. \(\cos \frac{y}{x} = \log_ex + c\)
• C. \(\cos \frac{x}{y} = \log_ey + c\)
• D. \(\cos \frac{y}{x} = \log_ey + c\)

Hint:

In differential equations, separating variables can simplify the integration process, especially when the equation allows for straightforward separation and substitution.

Answer 1

The Correct Option is B

Start by rearranging the given differential equation: \[ x \sin \left( \frac{y}{x} \right) \, dy = \left( y \sin \left( \frac{y}{x} \right) - x \right) \, dx. \] First, divide both sides of the equation by \(x \sin \left( \frac{y}{x} \right)\): \[ \frac{dy}{dx} = \frac{y \sin \left( \frac{y}{x} \right) - x}{x \sin \left( \frac{y}{x} \right)}. \] Simplify the equation by separating the terms: \[ \frac{dy}{dx} = \frac{y}{x} - \frac{x}{x \sin \left( \frac{y}{x} \right)}. \] Now, integrate both sides of the equation: \[ \int \frac{dy}{y} = \int \left( \frac{1}{x} - \frac{1}{x \sin \left( \frac{y}{x} \right)} \right) dx. \] On integrating, we get: \[ \log |y| = \log |x| + c. \] Thus, solving for the general solution, we obtain: \[ \cos \left( \frac{y}{x} \right) = \log_e x + c. \]