To solve Problem 2, let's analyze the system:

The equation of motion for the mass-spring-damper system is given as:

$m\ddot{x} + b\dot{x} + kx = 0$

Where:

• $m = 1$ (mass),
• $b = 2$ (damping coefficient),
• $k = 1$ (spring constant),
• $x(0) = 4$ (initial displacement),
• $\dot{x}(0) = 0$ (initial velocity).

Step 1: Characteristic Equation

The characteristic equation is obtained by substituting $x(t) = e^{\lambda t}$ into the differential equation:

$m\lambda^2 + b\lambda + k = 0$

Substitute $m = 1$, $b = 2$, and $k = 1$:

$\lambda^2 + 2\lambda + 1 = 0$

Factorize the equation:

$(\lambda + 1)^2 = 0$

So, $\lambda = -1$ (repeated root).

Step 2: General Solution

For a repeated root $\lambda = -1$, the general solution of the differential equation is:

$x(t) = (C_1 + C_2t)e^{\lambda t}$

Substitute $\lambda = -1$:

$x(t) = (C_1 + C_2t)e^{-t}$

Step 3: Apply Initial Conditions

1. Initial Condition 1: $x(0) = 4$:

Substitute $t = 0$ into $x(t)$:

$x(0) = C_1e^0 = C_1 = 4$

So, $C_1 = 4$.

2. Initial Condition 2: $\dot{x}(0) = 0$:

Differentiate $x(t)$:

$\dot{x}(t) = -e^{-t}(C_1 + C_2t) + e^{-t}C_2 = e^{-t}(-C_1 - C_2t + C_2)$

At $t = 0$:

$\dot{x}(0) = -C_1 + C_2 = 0$

Substitute $C_1 = 4$:

$-4 + C_2 = 0 \implies C_2 = 4$

Step 4: Final Solution

Substitute $C_1 = 4$ and $C_2 = 4$ into $x(t)$:

$x(t) = (4 + 4t)e^{-t}$

This is the motion of the system.

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Explanation of Steps

1. Derived the characteristic equation from the differential equation.
2. Solved for the roots of the characteristic equation, which are repeated.
3. Constructed the general solution for a repeated root.
4. Used the initial conditions to find the constants $C_1$ and $C_2$.
5. Substituted the constants back into the general solution to obtain the final expression.

Would you like a graph of this solution or further explanation?

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Related Questions

1. How does the damping coefficient $b$ affect the solution for different values?
2. What happens if the roots of the characteristic equation are complex instead of repeated?
3. How would the solution change if the initial velocity were not zero?
4. What is the significance of the exponential decay factor $e^{-t}$?
5. How would you analyze the system if external forces were acting on it?

Tip

For repeated roots, always check if your solution includes both terms $e^{\lambda t}$ and $te^{\lambda t}$, as the