The given recurrence relation is:

$a_n = 5a_{n-1} - 4a_{n-2} + n^2 \quad \text{with initial conditions} \quad a_1 = 1, \quad a_2 = 2$

To solve this recurrence relation, we proceed in two steps:

1. Solve the homogeneous recurrence relation $a_n = 5a_{n-1} - 4a_{n-2}$.
2. Find a particular solution to the non-homogeneous recurrence relation $a_n = 5a_{n-1} - 4a_{n-2} + n^2$.

Step 1: Solve the homogeneous recurrence relation

The homogeneous recurrence is: $a_n = 5a_{n-1} - 4a_{n-2}$

The characteristic equation for this recurrence relation is: $r^2 - 5r + 4 = 0$

Factoring this equation gives: $(r - 1)(r - 4) = 0$ So, the roots are $r_1 = 1$ and $r_2 = 4$.

Therefore, the general solution to the homogeneous equation is: $a_n^{(h)} = C_1 \cdot 1^n + C_2 \cdot 4^n = C_1 + C_2 \cdot 4^n$

Step 2: Find a particular solution

The non-homogeneous term is $n^2$, which suggests trying a particular solution of the form: $a_n^{(p)} = An^2 + Bn + C$ We substitute this into the original recurrence relation:

$An^2 + Bn + C = 5(A(n-1)^2 + B(n-1) + C) - 4(A(n-2)^2 + B(n-2) + C) + n^2$

Expanding and equating the coefficients of powers of $n$, we can solve for $A$, $B$, and $C$.

Step 3: Combine the general solution

The general solution to the recurrence relation is: $a_n = a_n^{(h)} + a_n^{(p)} = C_1 + C_2 \cdot 4^n + An^2 + Bn + C$

We then use the initial conditions $a_1 = 1$ and $a_2 = 2$ to solve for the constants $C_1$, $C_2$, $A$, $B$, and $C$.

Would you like me to calculate the values for the particular solution and constants next, or should I clarify any part so far?

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Follow-up Questions:

1. How do we use initial conditions to solve for constants in the general solution?
2. Can you explain why we use $An^2 + Bn + C$ for the particular solution?
3. What role does the characteristic equation play in solving recurrence relations?
4. How can we apply recurrence relations in solving algorithms and computational problems?
5. Why do non-homogeneous recurrence relations have both homogeneous and particular solutions?

Tip: Always check the degree of the non-homogeneous term to decide the form of the particular solution.