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This problem presents a second-order non-homogeneous linear differential equation with initial conditions. The equation is:

\[
x''(t) - 4x'(t) + 3x(t) = 4e^{3t}
\]
with the initial conditions:
\[
x(0) = 2, \quad x'(0) = 7
\]

### Steps to solve:
1. **Solve the homogeneous equation**:
   First, solve the homogeneous equation associated with the given differential equation:
   \[
   x''(t) - 4x'(t) + 3x(t) = 0
   \]
   The characteristic equation for this is:
   \[
   r^2 - 4r + 3 = 0
   \]
   Solve for \(r\) to find the general solution.

2. **Solve the particular solution**:
   Since the non-homogeneous term is \(4e^{3t}\), we'll assume a particular solution of the form:
   \[
   x_p(t) = Ae^{3t}
   \]
   Substitute this into the original equation to solve for \(A\).

3. **General solution**:
   The general solution is the sum of the homogeneous and particular solutions.

4. **Apply initial conditions**:
   Use \(x(0) = 2\) and \(x'(0) = 7\) to solve for the constants in the general solution.

Let me know if you'd like me to go through each step in detail, and if you'd like more clarification on any part of the process!

### Here are some related questions:
1. What is the characteristic equation of a general second-order linear differential equation?
2. How do you find the particular solution for non-homogeneous equations with exponential terms?
3. What is the role of initial conditions in solving differential equations?
4. How does one apply the method of undetermined coefficients?
5. Can you explain the superposition principle in the context of differential equations?

**Tip**: Always verify your solution by substituting it back into the original equation!

Solve the second-order differential equation: x''(t) - 4x'(t) + 3x(t) = 4e^{3t} with initial conditions x(0) = 2 and x'(0) = 7.

Learn how to solve the second-order non-homogeneous linear differential equation x''(t) - 4x'(t) + 3x(t) = 4e^{3t} with initial conditions x(0) = 2 and x'(0) = 7. This guide covers solving the homogeneous equation, finding the particular solution, and applying initial conditions.

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