GATE 2018 EC – Question 60

Engineering Mathematics · Differential Equations · 2 marks · Numerical answer

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

$$\frac{d^2y}{dt^2}=-\frac{dy}{dt}-\frac{5y}{4}.$$

The initial conditions are $y(0)=1$ and $\left.\frac{dy}{dt}\right|_{t=0}=0$. The position (accurate to two decimal places) of the particle at $t=\pi$ is ________.

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Correct answer: -0.22 to -0.20

Explanation

The characteristic equation $m^2+m+\frac54=0$ has roots $m=-\frac12\pm j$. So $y=e^{-t/2}(A\cos t+B\sin t)$, with $y(0)=1$ giving $A=1$ and $y'(0)=0$ giving $B=\frac12$. At $t=\pi$: $y=e^{-\pi/2}(-1)=-0.208\approx-0.21$.