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The given differential equation is:

\[
y'' - 4y' + 4y = e^{2x}
\]

This is a second-order linear non-homogeneous differential equation. To solve it, we first find the particular solution \( y^* \), and the form of \( y^* \) depends on the right-hand side of the equation, which is \( e^{2x} \).

### Steps to determine the form of the particular solution:

1. **Characteristic Equation of the Homogeneous Equation**:
   First, we solve the homogeneous version of the equation \( y'' - 4y' + 4y = 0 \). The characteristic equation is:
   \[
   r^2 - 4r + 4 = 0
   \]
   This can be factored as:
   \[
   (r - 2)^2 = 0
   \]
   So, the solution to the homogeneous equation is:
   \[
   y_h = (C_1 + C_2x)e^{2x}
   \]

2. **Form of the Particular Solution**:
   Since the homogeneous solution already includes \( e^{2x} \) and \( xe^{2x} \), we need to multiply by an additional factor of \( x^2 \) to find a particular solution. Therefore, the form of the particular solution must be:

   \[
   y^* = (Ax^2 + Bx + C)e^{2x}
   \]

   This matches **option 4**.

### Answer:
The correct form to search for the particular solution is **option 4**: 

\[
y^* = (Ax^2 + Bx + C)e^{2x}
\]

Would you like further details or explanation on solving non-homogeneous differential equations?

Here are five related questions:
1. How do we solve for the constants \( A \), \( B \), and \( C \) in the particular solution?
2. What is the complete solution to the differential equation?
3. What happens if the right-hand side is a different function, such as \( e^{3x} \)?
4. Why do we use the method of undetermined coefficients in this case?
5. How do we apply the variation of parameters to solve this type of equation?

**Tip:** For non-homogeneous differential equations, always consider the form of the complementary solution before choosing the form of the particular solution!

Partial solution y* of a linear non-homogeneous second-order differential equation with constant coefficients y'' - 4y' + 4y = e^(2x) needs to be found in the form of one of the following options.

Find the particular solution for the differential equation y'' - 4y' + 4y = e^(2x) using the method of undetermined coefficients. Learn how to solve the homogeneous equation and find the form of the particular solution, suited for advanced calculus students.

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