GATE 2018 EC Q.50: solution and answer

GATE 2018 · ECQ.50 · 2 marks · NAT

The position of a particle $y (t)$ is described by the differential equation:

\frac{d ^{2} y}{d t ^{2}} = - \frac{d y}{d t} - \frac{5 y}{4} .

The initial conditions are $y (0) = 1$ and $\frac{d y}{d t}_{t = 0} = 0$ . The position (accurate to two decimal places) of the particle at $t = π$ is ________.

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Answer and solution

ANSWER

$y (π) = - e^{- π /2} \approx - 0.21$ .

Official answer: -0.23 to -0.19Source: Official answer key, GATE 2018 EC (IIT Guwahati)

Move every term to one side: $y^{''} + y^{'} + \frac{5}{4} y = 0$ . This is a linear, homogeneous ODE with constant coefficients.
Auxiliary equation: $m^{2} + m + \frac{5}{4} = 0 \Rightarrow m = \frac{- 1 \pm 1 - 5}{2} = \frac{- 1 \pm 2 j}{2} = - \frac{1}{2} \pm j$ .
The roots are complex, $α \pm j β$ with $α = - \frac{1}{2}$ and $β = 1$ , so the general solution is $y (t) = e^{- t /2} (A cos t + B sin t)$ .
Apply $y (0) = 1$ : $e^{0} (A \cdot 1 + B \cdot 0) = A = 1$ .
Differentiate with the product rule: $y^{'} (t) = - \frac{1}{2} e^{- t /2} (A cos t + B sin t) + e^{- t /2} (- A sin t + B cos t) = e^{- t /2} [(B - \frac{A}{2}) cos t - (A + \frac{B}{2}) sin t]$ At $t = 0$ : $y^{'} (0) = B - \frac{A}{2} = 0 \Rightarrow B = \frac{1}{2}$ .
So the solution that meets both initial conditions is $y (t) = e^{- t /2} (cos t + \frac{1}{2} sin t)$ .
Evaluate at $t = π$ , using $cos π = - 1$ and $sin π = 0$ : $y (π) = e^{- π /2} (- 1 + 0) = - e^{- π /2} \approx - 0.2079 \approx - 0.21$

KEY IDEA

Write the equation as $y^{''} + y^{'} + \frac{5}{4} y = 0$ . Its auxiliary roots are $- \frac{1}{2} \pm j$ , so the motion is a decaying oscillation, $e^{- t /2} (A cos t + B sin t)$ . Use the two initial conditions to find $A$ and $B$ . At $t = π$ the sine term is zero, so only $A cos π e^{- π /2}$ is left.

WATCH OUT FOR

Setting $B = 0$ because $y^{'} (0) = 0$ . The decay factor $e^{- t /2}$ also contributes to the derivative, so $y^{'} (0) = B - \frac{A}{2}$ and $B = \frac{1}{2}$ . At $t = π$ this particular slip does no harm, because $sin π = 0$ . The same slip does change the answer at other times. Another mistake is copying the signs wrongly when moving terms across, for example writing $m^{2} - m + \frac{5}{4} = 0$ . That gives a growing solution and $y (π) = - e^{π /2} \approx - 4.81$ . Some candidates also forget the negative sign that $cos π = - 1$ brings in and enter $+ 0.21$ .

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GATE 2018 EC Q.50: solution and answer

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