GATE 2024 EE Q.45: solution and answer

Official answer: B, DSource: Official answer key, GATE 2024 EE (IISc Bengaluru)

1. Test for linearity: an ODE in $x(t)$ is linear when it can be written as $a_1(t)\frac{dx}{dt}+a_0(t)x=f(t)$. That means $x$ and its derivatives appear only to the first power, are never multiplied by each other, and never sit inside a nonlinear function. The coefficients may be any functions of $t$, the independent variable. The initial condition $x(0)=0$ has no bearing on linearity.

2. (A) $\frac{dx}{dt}+tx=t^2{e}^t$ has the standard form $x^{\prime }+P(t)x=Q(t)$ with $P(t)=t$ and $Q(t)=t^2{e}^t$. Both depend only on $t$, so the equation is linear.

3. (B) The term $x\frac{dx}{dt}$ multiplies the dependent variable by its own derivative, so it is a degree-2 term in $(x,x^{\prime })$. The equation is nonlinear. The term $\frac{1}{2}{e}^t$ is only a forcing term in $t$.

4. (C) Rewrite it as $-\sin t\frac{dx}{dt}+\cos tx=1$. The coefficients $-\sin t$ and $\cos t$ are functions of $t$ alone, and $x$ and $x^{\prime }$ each appear to the first power. The equation is linear, with variable coefficients.

5. (D) The derivative sits inside an exponential, $e^{dx/dt}$, so the equation cannot be put in the form $a_1(t)x^{\prime }+a_0(t)x=f(t)$. Superposition fails because $e^{a+b}\ne e^a+e^b$. Solving for the derivative gives the same conclusion: $\frac{dx}{dt}=\ln (1-x)$, which is nonlinear in $x$. The equation is nonlinear.

6. So the nonlinear equations are (B) and (D).