The image you've provided contains a differential equation, and the task is to find the transfer function of the system. The equation is:

$\frac{d^2y(t)}{dt^2} + 3 \frac{dy(t)}{dt} + 2 y(t) = \frac{dx(t)}{dt} - 4 x(t)$

Step 1: Take the Laplace Transform of both sides

• For the left-hand side, the Laplace transform of the terms:
  • $\frac{d^2y(t)}{dt^2}$ becomes $s^2Y(s)$
  • $\frac{dy(t)}{dt}$ becomes $sY(s)$
  • $y(t)$ becomes $Y(s)$

Thus, the Laplace transform of the left-hand side becomes: $s^2 Y(s) + 3s Y(s) + 2 Y(s)$ which can be factored as: $(s^2 + 3s + 2) Y(s)$

• For the right-hand side, the Laplace transform of the terms:
  • $\frac{dx(t)}{dt}$ becomes $s X(s)$
  • $x(t)$ becomes $X(s)$

Thus, the Laplace transform of the right-hand side becomes: $s X(s) - 4 X(s)$ which can be factored as: $(s - 4) X(s)$

Step 2: Form the equation in the $s$-domain

Equating both sides, we have: $(s^2 + 3s + 2) Y(s) = (s - 4) X(s)$

Step 3: Solve for the transfer function

The transfer function $H(s)$ is the ratio of the output $Y(s)$ to the input $X(s)$: $H(s) = \frac{Y(s)}{X(s)} = \frac{s - 4}{s^2 + 3s + 2}$

Thus, the transfer function of the system is: $H(s) = \frac{s - 4}{s^2 + 3s + 2}$

If you'd like a more detailed breakdown or have any questions, feel free to ask! Here are some related questions to deepen the understanding:

1. How do we interpret the transfer function in terms of system behavior?
2. What is the significance of the poles and zeros in the transfer function?
3. How do we find the stability of the system from the transfer function?
4. How would the system behave if the input were a step function?
5. What is the inverse Laplace transform of the transfer function?

Tip: A transfer function allows you to analyze a system’s output based on its input and is useful for determining system stability and frequency response.