Which of the following functions is a solution to the differential equation $y^{\prime }=2y$?

• A. $y=2x$
• B. $y=e^{2x}$
• C. $y=x^2$
• D. $y=e^x$

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the problem. The differential equation y' = 2y means that the derivative of the function y with respect to x is equal to 2 times the function y itself. We need to determine which of the given functions satisfies this equation.
2. Step 2: Recall the concept of exponential functions. Exponential functions often appear in differential equations of this form because their derivatives are proportional to the original function. Specifically, if y = e^{kx}, then y' = k * e^{kx}.
3. Step 3: Test each given function by calculating its derivative and checking if it satisfies the equation y' = 2y. For example, for y = e^{2x}, compute y' = d/dx(e^{2x}) = 2e^{2x}, which matches the equation y' = 2y.
4. Step 4: For the other functions, compute their derivatives and verify if they satisfy the equation. For y = 2x, y' = 2, which does not equal 2y. For y = x^2, y' = 2x, which also does not equal 2y. For y = e^{x}, y' = e^{x}, which does not equal 2y.
5. Step 5: Conclude that the only function that satisfies the differential equation y' = 2y is y = e^{2x}.