GATE 2026 EE Q.14: solution and answer

Official answer: ASource: Official answer key, GATE 2026 EE (IIT Guwahati)

1. Linearity test. A differential equation is linear if $y$ and each of its derivatives appear only to the first power, are never multiplied by each other, and never sit inside a nonlinear function. The coefficients may depend on the independent variable $t$ in any way, and the forcing term on the right may be any function of $t$.

2. In $t^2\frac{d^{2}y}{d{t}^2}+7t\frac{dy}{dt}+8ty=10\sin (t)$ the terms in $y$ are $\frac{d^{2}y}{d{t}^2}$, $\frac{dy}{dt}$ and $y$. Each appears to the first power and is multiplied only by a function of $t$ ($t^2$, $7t$, $8t$). There is no $y^2$, no $y\frac{dy}{dt}$ and no $\sin y$. The right-hand side $10\sin (t)$ contains no $y$ at all, so it is only a forcing term.

3. Superposition check. Write the operator $L[y]=t^2{y}^{\prime \prime }+7t{y}^{\prime }+8ty$. Because differentiation is linear and each term is a function of $t$ times $y$ or one of its derivatives, $L[a{y}_1+b{y}_2]=aL[y_1]+bL[y_2]$ for any constants $a,b$. So the equation is linear, which rules out (B).

4. Time-invariance test. A linear system described by a differential equation is time-invariant only when every coefficient is a constant. Here the coefficients $t^2$, $7t$ and $8t$ change with $t$, so the system is time-varying.

5. Shift check. Let input $x(t)$ give output $y(t)$, so $t^2{y}^{\prime \prime }(t)+7t{y}^{\prime }(t)+8ty(t)=x(t)$. Replace $t$ by $t-t_0$ and put $z(t)=y(t-t_0)$: $$(t-t_0{)}^2{z}^{\prime \prime }(t)+7(t-t_0)z^{\prime }(t)+8(t-t_0)z(t)=x(t-t_0)$$ But for $z(t)$ to be the system's response to the delayed input $x(t-t_0)$ it would have to satisfy $t^2{z}^{\prime \prime }+7t{z}^{\prime }+8tz=x(t-t_0)$. The coefficients differ for $t_0\ne 0$, so a delayed input does not in general give a delayed output. (C) is false.

6. Ordinary vs partial. There is only one independent variable, $t$, and every derivative is an ordinary derivative $\frac{d}{dt}$, with no $\partial$. The highest derivative is $\frac{d^{2}y}{d{t}^2}$, so it is a second-order ordinary differential equation, not a partial one. (D) is false.

7. The equation is linear and time-varying. Only (A) is correct.