We start by solving the ordinary differential equation:

$u_n'(t) = a_n u_n(t) + c_n(t)$

where

$a_n = -5(n+1/2)^2, \quad c_n(t) = \frac{6(-1)^n}{\pi (n+1/2)} \cos(6t).$

Since this is a first-order linear ODE, we look for a solution of the form:

$u_n(t) = A \cos(6t) + B \sin(6t) + C e^{a_n t}.$

Step 1: Find $A$ and $B$

We assume a particular solution of the form:

$u_n^{\text{part}}(t) = A \cos(6t) + B \sin(6t).$

Taking the derivative,

$u_n^{\text{part}'}(t) = -6 A \sin(6t) + 6 B \cos(6t).$

Substituting into the ODE,

$-6 A \sin(6t) + 6 B \cos(6t) = a_n(A \cos(6t) + B \sin(6t)) + c_n(t).$

Expanding,

$-6 A \sin(6t) + 6 B \cos(6t) = a_n A \cos(6t) + a_n B \sin(6t) + \frac{6(-1)^n}{\pi (n+1/2)} \cos(6t).$

Equating coefficients of $\cos(6t)$ and $\sin(6t)$, we get:

1. $6B = a_n A + \frac{6(-1)^n}{\pi (n+1/2)}$.
2. $-6A = a_n B$.

Solving for $A$ and $B$,

$A = \frac{-a_n}{6} B.$

Substituting this into the first equation:

$6B = a_n \left( \frac{-a_n}{6} B \right) + \frac{6(-1)^n}{\pi (n+1/2)}.$

$6B = \frac{-a_n^2}{6} B + \frac{6(-1)^n}{\pi (n+1/2)}.$

Multiplying everything by 6,

$36B + a_n^2 B = \frac{36(-1)^n}{\pi (n+1/2)}.$

$B(36 + a_n^2) = \frac{36(-1)^n}{\pi (n+1/2)}.$

$B = \frac{36(-1)^n}{\pi (n+1/2) (36 + a_n^2)}.$

Using $a_n = -5(n+1/2)^2$, we get:

$a_n^2 = 25(n+1/2)^4.$

$B = \frac{36(-1)^n}{\pi (n+1/2) (36 + 25(n+1/2)^4)}.$

$A = \frac{-a_n}{6} B = \frac{5(n+1/2)^2}{6} B.$

$A = \frac{30(n+1/2)^2(-1)^n}{\pi (n+1/2) 6 (36 + 25(n+1/2)^4)}.$

$A = \frac{5(n+1/2)(-1)^n}{\pi (36 + 25(n+1/2)^4)}.$

Step 2: Find $C$

The general solution also includes a homogeneous part:

$C e^{a_n t}.$

The initial condition $u_n(0) = 0$ gives:

$A \cos(0) + B \sin(0) + C e^{0} = 0.$

$A + C = 0.$

$C = -A.$

Final Answers:

$A = \frac{5(n+1/2)(-1)^n}{\pi (36 + 25(n+1/2)^4)}.$

$B = \frac{36(-1)^n}{\pi (n+1/2) (36 + 25(n+1/2)^4)}.$

$C = -A = \frac{-5(n+1/2)(-1)^n}{\pi (36 + 25(n+1/2)^4)}.$

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Do you need more details or a verification step?

Related Questions:

1. How can we generalize this solution method for different PDEs?
2. What happens if the boundary conditions are changed?
3. How do we interpret the role of eigenfunctions in this method?
4. Can we derive an explicit series solution for $u(x,t)$ using these coefficients?
5. What physical systems can be modeled by this PDE?

Tip: Always check boundary and initial conditions after solving a differential equation to ensure correctness!