The example you provided discusses a spring-mass system and solves for the motion of the mass after it is released. Here's a simplified explanation:

1. Physical Setup:

   • A spring with stiffness $k = 700 \, \text{N/m}$ is hanging vertically.
   • A mass of $M = 0.05 \, \text{kg}$ is attached to the spring.
   • When the mass is pulled down and released, it oscillates up and down.

2. Newton’s Second Law:

   • Using $F = M \frac{d^2x}{dt^2}$, the forces acting on the mass are:
     • The spring force: $-kx$, opposing displacement.
   • The net force gives the equation of motion: $M \frac{d^2x}{dt^2} = -kx.$

3. Second-Order Differential Equation:

   • Rearranging, we get: $\frac{d^2x}{dt^2} + \frac{k}{M} x = 0.$
   • Substituting $k = 700 \, \text{N/m}$ and $M = 0.05 \, \text{kg}$: $\frac{d^2x}{dt^2} + 100x = 0.$
   • This is a second-order linear differential equation describing simple harmonic motion.

4. Solving the Equation:

   • The characteristic equation is: $m^2 + 100 = 0 \implies m = \pm 10i.$
   • Complex roots indicate oscillatory motion. The general solution is: $x(t) = c_1 \cos(10t) + c_2 \sin(10t),$ where $c_1$ and $c_2$ are constants determined by initial conditions.

5. Initial Conditions:

   • At $t = 0$, the displacement $x(0) = 0.05 \, \text{m}$, and velocity $\dot{x}(0) = 0$:
     • Substituting these, the constants are found to be: $c_1 = 0.05, \quad c_2 = 0.$

6. Particular Solution:

   • The specific solution becomes: $x(t) = 0.05 \cos(10t).$

7. Resulting Motion:

   • The mass undergoes simple harmonic motion:
     • Amplitude: $0.05 \, \text{m}$.
     • Frequency: $1.59 \, \text{cycles/sec}$.
     • Period: $0.628 \, \text{sec}$.

In short:

The spring-mass system oscillates with a cosine function, showing periodic, smooth up-and-down motion due to the spring's restoring force.

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Do you want more details on any specific step? Here are some related questions:

1. What is the significance of the characteristic equation?
2. How do the initial conditions influence the solution?
3. Why does the motion follow a cosine function?
4. What determines the frequency and amplitude of oscillation?
5. Can damping or air resistance be included in this type of problem?

Tip: Always check units carefully when substituting values into formulas to avoid calculation errors!