Question

The particular solution of the differential equation \[ y \frac{dx}{dy} = x \log x \quad \text{at} \quad x = e \text{ and } y = 1 \] is

• A. \( e^{xy} = 2 \)
• B. \( x = e^y \)
• C. \( xy = 2 \)
• D. \( \log x = 2y \)

Hint:

When solving first-order differential equations, try separating variables and integrating both sides to find the general solution.

Answer 1

The Correct Option is B

Step 1: Solve the differential equation.
We are given the differential equation \( y \frac{dx}{dy} = x \log x \). To solve this, we separate variables and integrate both sides: \[ \frac{dx}{x \log x} = \frac{dy}{y}. \] After solving, we find that \( x = e^y \) satisfies the given initial conditions.

Step 2: Conclusion.
Thus, the particular solution is \( x = e^y \), corresponding to option (B).