Question

Let \( f: (0, \infty) \to \mathbb{R} \) be a function which is differentiable at all points of its domain and satisfies the condition \( x^2 f'(x) = 2f(x) + 3 \), with \( f(1) = 4 \). Then \( 2f(2) \) is equal to:

• A. \( 29 \)
• B. \( 39 \)
• C. \( 19 \)
• D. \( 23 \)

Hint:

For first-order linear differential equations, use the method of integrating factors to solve. Substitute the given initial condition to find the particular solution.

Answer 1

The Correct Option is A

We are given the differential equation \( x^2 f'(x) = 2f(x) + 3 \) and the initial condition \( f(1) = 4 \). To solve for \( f(x) \), we first divide both sides of the equation by \( x^2 \): \[ f'(x) = \frac{2f(x) + 3}{x^2}. \] We solve this first-order linear differential equation using the method of integrating factors. After solving, we substitute \( x = 2 \) and calculate \( 2f(2) \). 
Final Answer: \( 2f(2) = 29 \).