GATE 2018 EE – Question 43

Signals and Systems · Linear time invariant and causal systems · 2 marks · Multiple choice

Consider a system governed by the following equations

$$\frac{dx_1(t)}{dt}=x_2(t)-x_1(t),\qquad\frac{dx_2(t)}{dt}=x_1(t)-x_2(t).$$

The initial conditions are such that $x_1(0)<x_2(0)<\infty$. Let $x_{1f}=\lim_{t\to\infty}x_1(t)$ and $x_{2f}=\lim_{t\to\infty}x_2(t)$. Which one of the following is true?

$x_{1f}<x_{2f}<\infty$
$x_{2f}<x_{1f}<\infty$
$x_{1f}=x_{2f}<\infty$
$x_{1f}=x_{2f}=\infty$

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Correct answer: (C) $x_{1f}=x_{2f}<\infty$

Explanation

Adding the equations gives $\frac{d(x_1+x_2)}{dt}=0$, so the sum is constant, and subtracting gives $\frac{d(x_1-x_2)}{dt}=-2(x_1-x_2)$, so the difference decays to zero. So both tend to the same finite value, the average of the initial values.