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The given equation is a linear differential equation expressed as:

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

where \( D \) represents the differential operator \( \frac{d}{dx} \).

### Step-by-Step Solution:

1. **Solve the Homogeneous Equation**:
   First, solve the homogeneous equation corresponding to the left-hand side:
   \[
   (D^2 - 4D + 3)y = 0
   \]
   The characteristic equation is obtained by substituting \( D = r \), where \( r \) is the root of the equation:
   \[
   r^2 - 4r + 3 = 0
   \]
   Solving this quadratic equation gives:
   \[
   (r - 3)(r - 1) = 0
   \]
   Thus, \( r = 3 \) and \( r = 1 \).

   The general solution to the homogeneous equation is:
   \[
   y_h = c_1 e^{3x} + c_2 e^{x}
   \]
   where \( c_1 \) and \( c_2 \) are constants.

2. **Solve the Non-Homogeneous Equation (Particular Solution)**:
   The non-homogeneous term is \( x^3 e^{2x} \). Since the exponential term is \( e^{2x} \), we try a particular solution of the form:
   \[
   y_p = x^3 (Ax^3 + Bx^2 + Cx + D)e^{2x}
   \]
   We would substitute this into the non-homogeneous equation to solve for the constants \( A, B, C, D \).

3. **Full Solution**:
   The complete solution to the differential equation is the sum of the homogeneous and particular solutions:
   \[
   y = y_h + y_p
   \]
   Substituting the values of \( y_h \) and \( y_p \) will give the full solution.

Would you like the detailed calculation of the particular solution or have any specific questions on this? Here are 5 related questions that might help deepen your understanding:

1. How do we determine the form of the particular solution for different right-hand sides?
2. Can you explain how to solve a homogeneous differential equation in general?
3. What are the common types of non-homogeneous terms and their corresponding particular solutions?
4. How does the method of undetermined coefficients work for second-order differential equations?
5. What is the significance of the characteristic equation in solving differential equations?

**Tip**: Always double-check the particular solution's form to avoid overlap with the homogeneous solution when dealing with repeated roots or exponential functions.

Explore the step-by-step solution of the linear differential equation (D^2 - 4D + 3)y = x^3 e^{2x}. Learn about characteristic equations, homogeneous and non-homogeneous solutions, and apply the method of undetermined coefficients. Suitable for undergraduate students studying differential equations.

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