GATE 2026 IN Q.20: solution and answer

Official answer: CSource: Official answer key, GATE 2026 IN (IIT Guwahati)

1. Linearity test for a differential equation: the system is linear when the output $x(t)$ and each of its derivatives appears only to the first power, with no products of these terms and no nonlinear functions of them. The coefficients may depend on $t$. That only makes the system time-varying. It does not make it nonlinear.

2. Equation (i): $\frac{d^{2}x}{d{t}^2}+2t\frac{dx}{dt}+x=u$. The coefficient $2t$ multiplies $\frac{dx}{dt}$, which appears to the first power. Suppose $x_1$ solves the equation for input $u_1$ and $x_2$ solves it for $u_2$. Differentiation is linear, so substituting $a{x}_1+b{x}_2$ gives $a{u}_1+b{u}_2$ on the left. Superposition holds, so (i) is linear and time-varying.

3. Equation (ii): the term ${\left(\frac{dx}{dt}\right)}^2$ squares a derivative. Let $x_1$ solve the equation for $u_1$ and $x_2$ solve it for $u_2$. Substituting $x_1+x_2$ gives $${\left({\dot{x}}_1+{\dot{x}}_2\right)}^2+2({\dot{x}}_1+{\dot{x}}_2)+(x_1+x_2)=u_1+u_2+2{\dot{x}}_1{\dot{x}}_2,$$ which is not $u_1+u_2$ in general. Scaling also fails. Substituting $a{x}_1$ gives $a^2{\dot{x}}_1^2+2a{\dot{x}}_1+a{x}_1$, and this is not $a{u}_1=a{\dot{x}}_1^2+2a{\dot{x}}_1+a{x}_1$ unless $a=0$, $a=1$ or ${\dot{x}}_1\equiv 0$. So (ii) is nonlinear.

4. Equation (iii): $\frac{d^{2}x}{d{t}^2}+2\frac{dx}{dt}+x=u$ has constant coefficients and every term is first-degree in $x$ or its derivatives. So (iii) is linear and time-invariant.

5. Only (ii) fails the linearity test. The only equation that does not describe a linear dynamical system is (ii), which is option (C).