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 \(C\).

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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 \(k\).

Explore the solution to the differential equation y' = 2y with an initial condition y(0) = 3. Understand the concept of exponential growth, the role of the constant C in the solution, and how initial conditions affect differential equations. Suitable for Grades 11-12.

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