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  1. \(\frac{d^3y}{dt^3}+a_1\frac{d^2y}{dt^2}+a_2\frac{dy}{dt}+a_3y = b_3u + b_2\frac{du}{dt}+ b_1\frac{d^2u}{dt^2} \) (with initial rest conditions)
  2. \(y\left( t \right) = \mathop \smallint \limits_0^t {e^{\alpha \left( {t - \tau } \right)}}\beta u\left( \tau \right)d\tau \)
  3. y = au + b, b ≠ 0
  4. y = au
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1 Answers

Option 3 : y = au + b, b ≠ 0

Concept:

A system is said to be linear if it satisfies to properties:

Additivity:

If states that, if an input x1(t) produces output y1(t) and another input x2(t) also acting along produces output y2(t), then, when path inputs acting on the system simultaneously, produce output y1(t) + y2(t).

If x1(t) → y1(t)

x2(t) → y2(t)

then x1(t) + x2(t) → y1(t) + y2(t)

Homogeneity:

It states that if input is scaled by c, then output also scaled by same amount.

if x1(t) → y1(t)

then cx1(t) → cy1(t): c is any real or Imaginary number

Solution:

y = au + b

y(t) = au(t) + b

u(t) = u1(t) + u2(t)

y(t) = au1(t) + au2(t) + b

y(t) ≠ +

y(t) ≠ y1(t) + y2(t)

Hence the system is non linear

y = au + b is a nonlinear system

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