Question

For each of the systems described by the following differential equations, find the system transfer function:

d²y(t)/dt² + 11 dy(t)/dt + 24y(t)=5 dx(t)/dt + 3x(t)

Answer 1

The student's differential equation is converted to the s-domain using the Laplace transform to find the system transfer function, which is H(s) = (5s + 3) / (s² + 11s + 24).

Step-by-step explanation:

The student is asking for the system transfer function of a differential equation that describes a dynamic system. To find the transfer function, we use the Laplace transform to convert the time domain differential equation into the s-domain. Let L denote the Laplace transform and assume zero initial conditions.

First, let's take the Laplace transform of both sides of the given differential equation:

L{d²y(t)/dt²} + 11L{dy(t)/dt} + 24L{y(t)} = 5L{dx(t)/dt} + 3L{x(t)}

Under the assumption of zero initial conditions, the Laplace transform of the derivatives gives us:

s²Y(s) - sy(0) - dy(0)/dt + 11(sY(s) - y(0)) + 24Y(s) = 5(sX(s) - x(0)) + 3X(s)

Since y(0) = dy(0)/dt = x(0) = 0, we have:

s²Y(s) + 11sY(s) + 24Y(s) = 5sX(s) + 3X(s)

We can now solve for the transfer function Y(s)/X(s):

Transfer Function, H(s) = Y(s)/X(s) = (5s + 3) / (s² + 11s + 24)

Therefore, the system transfer function is H(s) = (5s + 3) / (s² + 11s + 24).