GATE 2023 CH – Question 31

Engineering Mathematics · Differential Equations: First order and higher order equations, Cauchy-Euler equations · 1 mark · Numerical answer

The position $x(t)$ of a particle, at constant $\omega$, is described by the equation $\frac{d^2x}{dt^2} = -\omega^2x$. The initial conditions are $x(t = 0) = 1$ and $\left.\frac{dx}{dt}\right|_{t=0} = 0$. The position of the particle at $t = (3\pi/\omega)$ is __________ (in integer).

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Correct answer: -1

Explanation

The solution is $x(t) = A\cos\omega t + B\sin\omega t$. The condition $x(0) = 1$ gives $A = 1$ and $\dot{x}(0) = 0$ gives $B = 0$, so $x = \cos\omega t$. At $\omega t = 3\pi$, $x = \cos 3\pi = -1$.