To find the transfer function for the given differential equation:

$m \ddot{x}(t) + b \dot{x}(t) + k x(t) = u(t) + 2 \dot{u}(t)$

We will proceed by applying the Laplace transform, which converts the time-domain differential equation into the s-domain, allowing us to solve for the transfer function.

Step 1: Apply the Laplace transform to each term.

• The Laplace transform of $\ddot{x}(t)$ is $s^2 X(s)$.
• The Laplace transform of $\dot{x}(t)$ is $s X(s)$.
• The Laplace transform of $x(t)$ is $X(s)$.
• The Laplace transform of $u(t)$ is $U(s)$.
• The Laplace transform of $\dot{u}(t)$ is $s U(s)$.

Step 2: Transform the equation.

Now, applying the Laplace transform to the equation:

$m (s^2 X(s)) + b (s X(s)) + k X(s) = U(s) + 2 (s U(s))$

Step 3: Rearrange the equation to solve for $\frac{X(s)}{U(s)}$, the transfer function.

$X(s) \left( m s^2 + b s + k \right) = U(s) \left( 1 + 2s \right)$

Now, divide both sides by $U(s)$ to find the transfer function:

$\frac{X(s)}{U(s)} = \frac{1 + 2s}{m s^2 + b s + k}$

This is the transfer function of the system.

Would you like to go over more details about the solution? Here are some follow-up questions you might be interested in:

1. How do the values of $m$, $b$, and $k$ affect the transfer function?
2. What would happen if we had a different type of input, like $u(t) = \sin(t)$?
3. How do you compute the inverse Laplace transform to get back to the time domain?
4. What physical systems could this equation represent?
5. How would you interpret the poles and zeros of this transfer function?

Tip: When solving for transfer functions, always ensure that the Laplace transform is applied correctly to all terms, especially with respect to initial conditions.