Question

Find the transfer function, G (s) = C (s)/R (s), corresponding to the differential equation

d³c d²c dc d²r dr
___ + 3 ___ + 7 ___ +5C = ____ + 4 ___ + 3r.

dt³ dt² dt dt² dt

Answer 1

To find the transfer function \( G(s) = \frac{C(s)}{R(s)} \) for the given differential equation, we'll first take the Laplace transform of the entire equation. Let \( C(s) \) and \( R(s) \) represent the Laplace transforms of \( c(t) \) and \( r(t) \), respectively.

The given differential equation is:

\[ \frac{d^3c}{dt^3} + 3\frac{d^2c}{dt^2} + 7\frac{dc}{dt} + 5c = \frac{d^2r}{dt^2} + 4\frac{dr}{dt} + 3r \]

Taking the Laplace transform of both sides:

\[ s^3C(s) - s^2c(0) - sc'(0) - c''(0) + 3s^2C(s) - 3sc(0) - 7C(s) + 5C(s) = s^2R(s) - sr(0) - r'(0) + 4sR(s) - 4r(0) + 3R(s) \]

Now, rearrange the terms and solve for \( \frac{C(s)}{R(s)} \) to get the transfer function \( G(s) \):

\[ G(s) = \frac{C(s)}{R(s)} = \frac{s^2 + 4s + 3}{s^3 + 3s^2 + 7s + 5} \]

So, the transfer function corresponding to the given differential equation is \( G(s) = \frac{s^2 + 4s + 3}{s^3 + 3s^2 + 7s + 5} \).

Answer 2

To find the transfer function G(s) = C(s)/R(s), the Laplace transform is applied to the differential equation given by the student, resulting in a ratio of polynomials in 's', which represents the transfer function.

Step-by-step explanation:

The student's question is related to finding the transfer function G(s) = C(s)/R(s) from a given differential equation. To derive this transfer function, we apply the Laplace transform to both sides of the differential equation, taking into consideration that the Laplace transform of a derivative is sn multiplied by the Laplace transform of the function itself, minus the initial conditions (which are not given and therefore assumed to be zero). The detailed equation given is:

d3c/dt3 + 3 d2c/dt2 + 7 dc/dt +5c = d2r/dt2 + 4 dr/dt + 3r.

Applying the Laplace transform to each term and assuming the initial conditions are zero (since they are not provided), we get:

s3C(s) + 3s2C(s) + 7sC(s) + 5C(s) = s2R(s) + 4sR(s) + 3R(s).

Rearranging for C(s)/R(s), we obtain the transfer function:

G(s) = (s2 + 4s + 3) / (s3 + 3s2 + 7s + 5).