The plot above shows several solutions to the differential equation $y' = 2y$ with different values of $C$. The red dashed line represents the specific solution $y(x) = 3e^{2x}$ that satisfies the initial condition $y(0) = 3$. Each curve represents different exponential growth or decay based on the constant $C$.

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the general behavior of solutions to the differential equation $y' = ky$ for different values of $k$?
2. How does changing the initial condition $y(0) = y_0$ affect the solution?
3. What is the significance of the constant $C$ in the general solution $y(x) = Ce^{2x}$?
4. How would you solve a differential equation of the form $y' = ay + b$?
5. Can the differential equation $y' = 2y$ be solved using separation of variables? How?

Tip: For differential equations of the form $y' = ky$, solutions will exhibit exponential growth or decay depending on the sign of $k$.