The plot above shows several solutions to the differential equation $y' + y = 0$ with different values of $C$. The red dashed line represents the specific solution $y(x) = 2e^{-x}$ that satisfies the initial condition $y(0) = 2$. Each curve represents a different exponential decay based on the constant $C$.

Would you like more details or have any questions?

Here are some related questions to explore:

1. What happens to the solution as $x$ approaches infinity?
2. How would the solutions change if the differential equation were $y' - y = 0$?
3. What are the general characteristics of solutions to first-order linear differential equations?
4. How would you solve a first-order differential equation with a non-zero constant on the right-hand side, like $y' + y = 1$?
5. What methods can be used to solve first-order non-linear differential equations?

Tip: When solving differential equations, always verify your solution by substituting it back into the original equation to ensure it satisfies the equation.