Question

Find the general solution of the differential equation: $$ x \log x \, dy = (x \log x - y) dx $$

• A. \((x - y) \log x + x = c\)
• B. \(x - y = \frac{x}{\log x} + c\)
• C. \(y - x = \frac{x}{\log x} + c\)
• D. \((y - x) \log x + x = c\)

Hint:

Rewrite in standard form and solve using exact equation or integrating factor.

Answer 1

The Correct Option is D

Rewrite: \[ x \log x \, dy = (x \log x - y) dx \implies x \log x \, dy + y \, dx = x \log x \, dx \] Rearranged as: \[ M dx + N dy = 0 \] with \[ M = y - x \log x, \quad N = - x \log x \] Check for exactness or use integrating factor. Solve the differential equation to get: \[ (y - x) \log x + x = c \]