Question

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y, \] satisfying \( \varphi(1) = e^2 \). Then, the value of \( \varphi(2) \) is equal to:

• A. \( e^2 \)
• B. \( 2e^3 \)
• C. \( 3e^2 \)
• D. \( 6e^3 \)

Hint:

When solving a first-order differential equation, try separating the variables and integrating each side. Use the initial conditions to find the constant of integration.

Answer 1

The Correct Option is B

Step 1: Simplify the differential equation.
The given differential equation is: \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y \] Rewrite this as: \[ \frac{dy}{dx} = \left( \ln y - \ln x \right) \frac{y}{x} \] This simplifies to: \[ \frac{dy}{dx} = \left( \ln \frac{y}{x} \right) \frac{y}{x} \] Step 2: Separation of variables.
Separate variables to integrate: \[ \frac{dy}{y} = \ln \frac{y}{x} \, \frac{dx}{x} \] Integrating both sides: \[ \int \frac{1}{y} \, dy = \int \ln \frac{y}{x} \, \frac{1}{x} \, dx \] Step 3: Use initial conditions.
We are given that \( \varphi(1) = e^2 \), which provides the necessary condition to solve for the constants. After solving, the final solution is: \[ \varphi(2) = 2e^3 \] Final Answer: \[ \boxed{2e^3} \]