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The given equation is a differential equation, which appears as:

\[
\frac{d^3 y}{d c^3} + 2 \frac{d^2 y}{d c^2} + \frac{d y}{d c} = e^{2c} + \sin(3c)
\]

This is a linear non-homogeneous third-order differential equation. The solution to such equations typically involves finding two parts:
1. **The complementary (or homogeneous) solution**, where the right-hand side is set to zero.
2. **The particular solution**, which accounts for the non-homogeneous terms on the right-hand side.

### Steps to solve:
1. **Solve the homogeneous equation:**
   \[
   \frac{d^3 y}{d c^3} + 2 \frac{d^2 y}{d c^2} + \frac{d y}{d c} = 0
   \]
   Find the characteristic equation and its roots to obtain the complementary solution.

2. **Find the particular integral:**
   For the right-hand side, \( e^{2c} + \sin(3c) \), assume a particular solution that matches the form of each term:
   - For \( e^{2c} \), assume a solution of the form \( A e^{2c} \).
   - For \( \sin(3c) \), assume a solution of the form \( B \cos(3c) + C \sin(3c) \).

3. **Add both solutions (complementary + particular)** to get the general solution.

Would you like to proceed with the detailed solution for either or both parts? Let me know if you have specific queries regarding this.

Here are some follow-up questions related to this problem:

1. Would you like me to solve the homogeneous equation in detail?
2. Do you need a step-by-step approach to finding the particular integral?
3. Would you prefer an explanation of the method for solving third-order differential equations?
4. Are there any specific boundary conditions given for this problem?
5. Would you like to know how to apply initial conditions to find the complete solution?

**Tip:** When solving non-homogeneous differential equations, always handle the homogeneous and particular solutions separately before combining them for the full solution.

Particular integral of the differential equation: d^3y/dc^3 + 2d^2y/dc^2 + dy/dc = e^(2c) + sin(3c)

Explore how to find the particular integral of the third-order non-homogeneous differential equation: d^3y/dc^3 + 2d^2y/dc^2 + dy/dc = e^(2c) + sin(3c). Learn the step-by-step method of undetermined coefficients, suitable for Grades 11-12 or early university-level math students.

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