To solve the given differential equation dy/dx + 2y = 0, we can use the method of separation of variables. This method involves separating the variables dy and dx and then integrating both sides of the equation.

Separation of Variables:
1. Start by isolating the dy term on one side of the equation:
dy = -2y dx

2. Now, divide both sides of the equation by y:
dy/y = -2 dx

Integration:
3. Integrate both sides of the equation with respect to their respective variables:
∫ (1/y) dy = ∫ (-2) dx

4. The integral of (1/y) with respect to y is ln|y| + C1, where C1 is the constant of integration.
The integral of -2 with respect to x is -2x + C2, where C2 is another constant of integration.

5. Therefore, we have the equation:
ln|y| + C1 = -2x + C2

6. Combining the constants of integration, we can write the equation as:
ln|y| = -2x + C, where C = C2 - C1

Exponential Function:
7. To eliminate the natural logarithm, we can take the exponential of both sides of the equation:
e^(ln|y|) = e^(-2x + C)

8. By the properties of logarithms, the exponential of ln|y| simplifies to |y|:
|y| = e^(-2x + C)

9. Since e^C is just another constant, we can rewrite the equation as:
|y| = Ce^(-2x)

10. Finally, we can express the absolute value as a positive constant:
y = Ce^(-2x), where C is a constant.

Conclusion:
The function y = Ce^(-2x) is the solution to the given differential equation dy/dx + 2y = 0. Therefore, option C is the correct answer.