This solution outlines the setup and analysis of a forced, undamped harmonic oscillator. Let’s break down the key parts of the process:

1. Establishing the Standard Form: The problem is set up using the differential equation $m u'' + \gamma u' + k u = F(t)$, where:

   • $m$ is the mass,
   • $\gamma$ is the damping coefficient,
   • $k$ is the spring constant, and
   • $F(t)$ is the external forcing function.

2. Given Data and Unit Conversions:

   • The mass $mg = 13$ lb, which is converted to slugs as $m = \frac{13}{32} = 0.40625$ slugs (using $g = 32 \, \text{ft/s}^2$).
   • The spring constant $k$ is calculated using $k s = 13$ and the stretch $s = \frac{8}{12} = \frac{2}{3}$ ft. This gives $k = \frac{13}{(2/3)} = 19.5 \, \text{lb/ft}$.

3. Damping and Forcing:

   • Since there’s no damping, $\gamma = 0$.
   • The external force is given as $F(t) = 7 \sin(4 \sqrt{3} \, t) \, \text{lb}$.

4. Initial Conditions:

   • The mass is pulled down 3 inches, giving an initial displacement of $u(0) = \frac{3}{4} = 0.75 \, \text{ft}$.
   • The initial velocity is zero, so $u'(0) = 0$.

5. Simplifying the Differential Equation:

   • The differential equation becomes $0.40625 u'' + 19.5 u = 7 \sin(4 \sqrt{3} \, t)$.
   • Dividing through by $0.40625$ simplifies it to $u'' + 48 u = \frac{224}{13} \sin(4 \sqrt{3} \, t)$.

6. Characteristic Equation:

   • The characteristic equation for the homogeneous system $u'' + 48 u = 0$ is $\lambda^2 + 48 = 0$, leading to solutions in terms of imaginary roots, indicating oscillatory motion.

Would you like a more detailed analysis on a specific part of this solution or explanations of any steps?

Here are 5 related questions that could deepen your understanding:

1. How would the solution change if there was a damping coefficient $\gamma \neq 0$?
2. What are the general solutions for the homogeneous equation $u'' + 48u = 0$?
3. How does the external forcing function $F(t) = 7 \sin(4 \sqrt{3} \, t)$ affect the system's steady-state behavior?
4. Can we determine the amplitude of the resulting oscillation from this solution?
5. What role does the initial displacement $u(0) = 0.75$ ft play in the solution?

Tip: When solving differential equations with sinusoidal forcing functions, look out for resonance conditions, which occur when the forcing frequency matches the natural frequency of the system.