GATE 2025 EE – Question 54

Engineering Mathematics · Differential Equations: First order equations, Higher order linear differential equations, Variation of parameters, Cauchy and Euler equations, Partial differential equations, Separation of variables · 2 marks · Numerical answer

Consider ordinary differential equations given by

$\dot x_1(t)=2x_2(t)$

$\dot x_2(t)=r(t)$

with initial conditions $x_1(0)=1$ and $x_2(0)=0$.

If $r(t)=\begin{cases}1,&t\ge0\\0,&t<0\end{cases}$, then at $t=1$, $x_1(t)=$ __________ (round off to the nearest integer).

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Correct answer: 2

Explanation

$x_2(t)=\int_0^t1\,d\tau=t$. Then $x_1(t)=1+\int_0^t2\tau\,d\tau=1+t^2$. At $t=1$, $x_1=2$.