Question

If \( 2f(x) = f'(x) \) and \( f(0) = 3 \), then the value of \( f(2) \) is

• A. \( 3e^2 \)
• B. \( 2e^3 \)
• C. \( 4e^3 \)
• D. \( 3e^4 \)

Hint:

For differential equations of the form \( f'(x) = kf(x) \), solve by separating variables and applying the given initial conditions to find the solution.

Answer 1

The Correct Option is D

Step 1: Solve the differential equation.
We are given that \( 2f(x) = f'(x) \). This is a first-order linear differential equation. Rearranging the terms, we get: \[ \frac{f'(x)}{f(x)} = 2 \] Integrating both sides: \[ \int \frac{f'(x)}{f(x)} \, dx = \int 2 \, dx \] This gives: \[ \ln |f(x)| = 2x + C \] Thus: \[ f(x) = e^{2x + C} = e^C e^{2x} \]
Step 2: Apply the initial condition.
We are also given that \( f(0) = 3 \). Substituting \( x = 0 \) into the equation for \( f(x) \), we get: \[ f(0) = e^C = 3 \] Thus, \( e^C = 3 \), so \( C = \ln 3 \).
Step 3: Find \( f(2) \).
Now that we know \( f(x) = 3e^{2x} \), we substitute \( x = 2 \) into the expression for \( f(x) \): \[ f(2) = 3e^{4} \]
Step 4: Conclusion.
Thus, the value of \( f(2) \) is \( 3e^4 \).