Solve the differential equation $\frac{dy}{dx}=2y$ by separation of variables. Which of the following is the general solution?

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

Answer: (A)

Verified step by step guidance

1. Step 1: Start by recognizing that the differential equation dy/dx = 2y is separable, meaning you can rewrite it to separate the variables y and x.
2. Step 2: Rewrite the equation as (1/y) dy = 2 dx. This separates the variables y and x, allowing us to integrate each side independently.
3. Step 3: Integrate both sides. The left-hand side ∫(1/y) dy integrates to ln|y|, and the right-hand side ∫2 dx integrates to 2x + C₁, where C₁ is the constant of integration.
4. Step 4: Solve for y by exponentiating both sides to remove the natural logarithm. This gives |y| = e^(2x + C₁). Simplify further by writing |y| = Ce^(2x), where C = e^(C₁) is a new constant.
5. Step 5: Remove the absolute value by considering both positive and negative values of C, resulting in the general solution y = Ce^(2x).