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GATE 2026 IN – Question 20

Signals and Systems · Signals and linear time-invariant systems · 1 mark · Multiple choice

Among the following, the differential equation(s) not representative of a linear dynamical system is/are ________.

(i) $\frac{d^2x}{dt^2}+2t\frac{dx}{dt}+x(t)=u(t)$

(ii) $\left(\frac{dx}{dt}\right)^2+2\frac{dx}{dt}+x(t)=u(t)$

(iii) $\frac{d^2x}{dt^2}+2\frac{dx}{dt}+x(t)=u(t)$

  1. (i) and (ii) only
  2. (i) only
  3. (ii) only
  4. (ii) and (iii) only

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Correct answer: (C) (ii) only

Explanation

A linear differential equation contains the dependent variable and its derivatives only to the first power, without products between them. Its coefficients may depend on time.

Equation (i) is linear with a time-varying coefficient. Equation (iii) is linear with constant coefficients. Equation (ii) contains the square of a derivative, which violates superposition.

Only **(ii), option C**, is nonlinear.