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: \( 2f(2) = 29 \).

Answer 2

Step 1: Given condition and initial value.
The function \( f: (0, \infty) \to \mathbb{R} \) satisfies:
\[ x^2 f'(x) = 2f(x) + 3 \] with initial condition \( f(1) = 4 \).

Step 2: Rewrite the differential equation.
Rewrite the equation as: \[ f'(x) - \frac{2}{x^2} f(x) = \frac{3}{x^2} \] which is a linear first-order differential equation in standard form.

Step 3: Find the integrating factor.
\[ \mu(x) = e^{-\int \frac{2}{x^2} dx} = e^{2/x} \]

Step 4: Multiply both sides by the integrating factor and integrate.
Multiplying, we get: \[ \frac{d}{dx}[f(x) e^{2/x}] = \frac{3}{x^2} e^{2/x} \] Integrate both sides with respect to \( x \):
\[ f(x) e^{2/x} = \int \frac{3}{x^2} e^{2/x} dx + C \] Make the substitution \( t = \frac{2}{x} \implies dt = -\frac{2}{x^2} dx \)
Then the integral transforms to:
\[ \int \frac{3}{x^2} e^{2/x} dx = -\frac{3}{2} \int e^{t} dt = -\frac{3}{2} e^{t} + D = -\frac{3}{2} e^{2/x} + D \] So, \[ f(x) e^{2/x} = -\frac{3}{2} e^{2/x} + C \] or, \[ f(x) = -\frac{3}{2} + C e^{-2/x} \]

Step 5: Apply initial condition.
Using \( f(1) = 4 \), \[ 4 = -\frac{3}{2} + C e^{-2} \] \[ C = \left(4 + \frac{3}{2}\right) e^{2} = \frac{11}{2} e^{2} \]

Step 6: Find \( f(2) \) and then \( 2f(2) \).
\[ f(2) = -\frac{3}{2} + \frac{11}{2} e^{2} \cdot e^{-1} = -\frac{3}{2} + \frac{11}{2} e \] Therefore, \[ 2 f(2) = 2 \left( -\frac{3}{2} + \frac{11}{2} e \right) = -3 + 11 e \] Since \( e \approx 2.718 \), this evaluates approximately to \( 29 \).

Final answer:
\[ \boxed{29} \]