A second order system is governed by y¨(t)+2y˙(t)=u(t)𝑦¨(𝑡)+2𝑦˙(𝑡)=𝑢(𝑡). Its transfer function is1s2+21𝑠2+21s2−21𝑠2−21s2+2s1𝑠2+2𝑠1s2−2s

It seems like there's a bit of confusion in your question as the transfer function you've provided appears to be a bit jumbled. However, I'll provide a general way to find the transfer function of a second order system.

A second order system is typically represented by the following differential equation:

y''(t) + ay'(t) + by(t) = u(t)

where:

• y''(t) is the second derivative of the output with respect to time,
• y'(t) is the first derivative of the output with respect to time,
• y(t) is the output,
• u(t) is the input,
• a and b are constants.

The Laplace transform of this equation is:

s^2Y(s) + asY(s) + bY(s) = U(s)

where:

• Y(s) is the Laplace transform of y(t),
• U(s) is the Laplace transform of u(t),
• s is the complex frequency.

The transfer function, G(s), is the ratio of the output, Y(s), to the input, U(s). So, we can rearrange the above equation to find G(s):

G(s) = Y(s)/U(s) = 1/(s^2 + a*s + b)

So, for your system where a = 2 and there's no y(t) term, the transfer function should be:

G(s) = 1/(s^2 + 2s)