Question

Let \( x = x(y) \) be the solution of the differential equation} \[ 2y^2 \frac{dx}{dy} - 2xy + x^2 = 0, \quad y>1, \quad x(e) = e. \] Then \( x(e^2) \) is equal to:

• A. \( \frac{3}{2} e^2 \)
• B. \( \frac{2}{3} e^2 \)
• C. \( e^2 \)
• D. \( 2e^2 \)

Answer 1

The Correct Option is C

We are given the differential equation: \[ 2y^2 \frac{dx}{dy} - 2xy + x^2 = 0. \] To solve this, first rearrange the equation: \[ 2y^2 \frac{dx}{dy} = 2xy - x^2. \] Now, divide both sides by \( 2y^2 \) to isolate \( \frac{dx}{dy} \): \[ \frac{dx}{dy} = \frac{x(2y - x)}{2y^2}. \] This is a separable differential equation. We separate the variables: \[ \frac{dx}{x(2y - x)} = \frac{dy}{2y^2}. \] Next, solve the left-hand side and right-hand side separately. After integration and solving for \( x \), use the initial condition \( x(e) = e \) to find the constant of integration. The final solution is: \[ x(e^2) = e^2. \]
Final Answer: \( e^2 \)