Question

Convert between differential equation and Laplace Transform. a) The following differential equation describes a linear system's output y(t) vs. input x(t). Find the system's transfer function H(s) 2 d²y(t)/dt²​+dy(t)/dt​+y(t)=d²x(t)​/dt²+dx(t)​/dt+2x(t)

Answer 1

To find the transfer function H(s) of the given differential equation, apply the Laplace Transform to both sides of the equation, simplify and solve for H(s), and compare the resulting equation to known transfer functions.

Step-by-step explanation:

To find the transfer function H(s) of the linear system described by the given differential equation, we need to apply the Laplace Transform. The Laplace Transform is a mathematical technique used to convert a time-domain differential equation into the frequency-domain. In this case, we will apply the Laplace Transform to both sides of the differential equation.

1. Apply the Laplace Transform to each term of the differential equation. The Laplace Transform of the derivative terms can be found using the property: L{f '(t)} = sF(s) - f(0), where F(s) is the Laplace Transform of f(t) and f(0) represents the initial condition.
2. Simplify and solve for H(s) by rearranging the equation. The goal is to isolate H(s) on one side of the equation.
3. Observe the resulting equation and compare it to known transfer functions to determine the system's transfer function H(s). In this case, the given equation has a similar form to RC circuits, so we can make the connection to find the transfer function H(s). The transfer function represents the relationship between the input and output of a system in the frequency domain.