To solve the first differential equation:

\[ \ddot{y} - 2\dot{y} = 3e^{2x}, \]

we'll follow these steps:

### Step 1: Solve the Homogeneous Equation

First, solve the homogeneous part of the equation:

\[ \ddot{y} - 2\dot{y} = 0. \]

This is a second-order linear differential equation with constant coefficients. The characteristic equation is:

\[ r^2 - 2r = 0. \]

Factoring out \( r \):

\[ r(r - 2) = 0. \]

So, the roots are:

\[ r = 0 \quad \text{and} \quad r = 2. \]

The general solution to the homogeneous equation is:

\[ y_h(x) = C_1 e^{0 \cdot x} + C_2 e^{2x} = C_1 + C_2 e^{2x}, \]

where \( C_1 \) and \( C_2 \) are arbitrary constants.

### Step 2: Find a Particular Solution

Now, let's find a particular solution to the non-homogeneous equation:

\[ \ddot{y} - 2\dot{y} = 3e^{2x}. \]

Since the right-hand side is \( 3e^{2x} \), and it's already a solution form of the homogeneous equation, we try a particular solution of the form:

\[ y_p(x) = Ax e^{2x}. \]

Differentiate this twice:

\[ \dot{y}_p(x) = A(e^{2x} + 2xe^{2x}) = A(2x + 1)e^{2x}, \]
\[ \ddot{y}_p(x) = A(4x + 4)e^{2x}. \]

Substitute \( y_p(x) \), \( \dot{y}_p(x) \), and \( \ddot{y}_p(x) \) into the original equation:

\[ A(4x + 4)e^{2x} - 2A(2x + 1)e^{2x} = 3e^{2x}. \]

Simplifying, we get:

\[ (4A x + 4A - 4A x - 2A)e^{2x} = 3e^{2x}, \]

which simplifies to:

\[ 2A e^{2x} = 3e^{2x}. \]

Thus, \( A = \frac{3}{2} \).

So the particular solution is:

\[ y_p(x) = \frac{3}{2} x e^{2x}. \]

### Step 3: Write the General Solution

The general solution to the differential equation is the sum of the homogeneous and particular solutions:

\[ y(x) = y_h(x) + y_p(x). \]

Thus,

\[ y(x) = C_1 + C_2 e^{2x} + \frac{3}{2} x e^{2x}. \]

This is the general solution to the given differential equation.

Would you like further details or explanations on any step? Here are some related questions:

1. How do you determine the form of a particular solution for non-homogeneous differential equations?
2. What is the significance of the characteristic equation in solving differential equations?
3. How can you verify that a solution satisfies a differential equation?
4. What are some common methods for solving second-order differential equations?
5. How do initial conditions affect the constants in the general solution of a differential equation?

**Tip:** When finding a particular solution, ensure that it doesn't overlap with the homogeneous solution; otherwise, multiply by \( x \) to adjust it.

Learn how to solve the second-order differential equation y'' - 2y' = 3e^(2x) step-by-step with detailed explanations. This problem covers differential equations, including homogeneous equations and particular solutions. Suitable for advanced undergraduate level.

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