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:
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For first-order linear differential equations, use the method of integrating factors to solve. Substitute the given initial condition to find the particular solution.
Given the differential equation \( x^2 f'(x) = 2f(x) + 3 \) and the initial condition \( f(1) = 4 \), we seek to determine \( f(x) \). The equation is first rewritten as:
\[
f'(x) = \frac{2f(x) + 3}{x^2}.
\]
This first-order linear differential equation is solved using the method of integrating factors. Following the solution, we will substitute \( x = 2 \) to compute \( 2f(2) \).
