GATE 2018 EC Q.4: solution and answer

GATE 2018 · ECQ.4 · 1 mark · MCQ

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:

(A)
$\frac{d ^{3} y}{d t ^{3}} + a_{1} \frac{d ^{2} y}{d t ^{2}} + a_{2} \frac{d y}{d t} + a_{3} y = b_{3} u + b_{2} \frac{d u}{d t} + b_{1} \frac{d ^{2} u}{d t ^{2}}$ (with initial rest conditions)
(B)
$y (t) = \int_{0}^{t} e^{α (t - τ)} β u (τ) d τ$
(C)
$y = a u + b, b \neq = 0$
(D)
$y = a u$

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Answer and solution

ANSWER

(C) $y = a u + b$ with $b \neq = 0$ is not linear. It gives $y = b \neq = 0$ when $u = 0$ , so superposition fails. (A), (B) and (D) all satisfy superposition.

Official answer: CSource: Official answer key, GATE 2018 EC (IIT Guwahati)

The test is superposition. A system is linear only if it is additive and homogeneous: an input $k_{1} u_{1} + k_{2} u_{2}$ must give the output $k_{1} y_{1} + k_{2} y_{2}$ . Setting $k_{1} = k_{2} = 0$ gives a quick necessary check: zero input must give zero output.
(A) This is a linear ODE with constant coefficients. Every term is first-degree in $y$ , $u$ or one of their derivatives, and no term multiplies two of them together. Initial rest means all initial conditions are zero, so there is no stored-energy (zero-input) term, and the output depends linearly on the input. So (A) is linear.
(B) This is a convolution integral, $y (t) = \int_{0}^{t} h (t - τ) u (τ) d τ$ , with kernel $h (t - τ) = β e^{α (t - τ)}$ . The kernel does not depend on $u$ . Integration is linear, so the input $k_{1} u_{1} + k_{2} u_{2}$ gives $k_{1} y_{1} + k_{2} y_{2}$ . So (B) is linear.
(D) $y = a u$ is a pure gain: $a (k_{1} u_{1} + k_{2} u_{2}) = k_{1} (a u_{1}) + k_{2} (a u_{2})$ . So (D) is linear.
(C), zero-input test. Put $u = 0$ . Then $y = b \neq = 0$ . A linear system must give $y = 0$ for zero input, so (C) fails.
(C), additivity check. Input $u_{1}$ gives $y_{1} = a u_{1} + b$ and input $u_{2}$ gives $y_{2} = a u_{2} + b$ . Input $u_{1} + u_{2}$ gives $a (u_{1} + u_{2}) + b \neq = y_{1} + y_{2} = a (u_{1} + u_{2}) + 2 b (since b \neq = 0) .$ Superposition fails. The graph of $y$ against $u$ is a straight line, but it does not pass through the origin, so the map is affine, not linear.
(C) is the only system that is not linear. Answer: (C).

KEY IDEA

A linear system must obey superposition (additivity plus homogeneity). A necessary consequence is that zero input gives zero output. A constant offset $b \neq = 0$ breaks this, so $y = a u + b$ is affine, not linear. A constant-coefficient ODE at initial rest and a convolution integral are both linear.

WATCH OUT FOR

Calling $y = a u + b$ linear because its graph is a straight line. A straight line through the origin is linear; any other straight line is affine. A second trap is doubting (A) because it contains derivatives of $u$ . Derivatives are linear operators, and initial rest removes any zero-input response, so (A) is linear.

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GATE 2018 EC Q.4: solution and answer

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