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.
The differential equation \( x^2 f'(x) = 2f(x) + 3 \) with the initial condition \( f(1) = 4 \) is provided. To find \( f(x) \), the equation is first rewritten as \( f'(x) = \frac{2f(x) + 3}{x^2} \) by dividing both sides by \( x^2 \). This first-order linear differential equation is solved using integrating factors. Subsequently, \( x = 2 \) is substituted, and \( 2f(2) \) is calculated. Final Answer: \( 2f(2) = 29 \).
