The GATE Grind

GATE 2018 EC – Question 14

Networks, Signals and Systems · LTI Systems · 1 mark · Multiple choice

Let the input be $u$ and the output be $y$ of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:

  1. $\dfrac{d^3y}{dt^3}+a_1\dfrac{d^2y}{dt^2}+a_2\dfrac{dy}{dt}+a_3y=b_3u+b_2\dfrac{du}{dt}+b_1\dfrac{d^2u}{dt^2}$ (with initial rest conditions)
  2. $y(t)=\displaystyle\int_0^te^{\alpha(t-\tau)}\beta u(\tau)\,d\tau$
  3. $y=au+b,\quad b\ne0$
  4. $y=au$

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Correct answer: (C) $y=au+b,\quad b\ne0$

Explanation

A linear system must give zero output for zero input. In C, $u=0$ gives $y=b\ne0$, so superposition fails. The others are linear (a linear ODE with zero initial conditions, a convolution, and a pure gain).