Consider the differential equation (t2 - 81)dydt + 5ty = (t) with y(1) = 2. There exists a unique solution for this differential equation when t belongs to the interval

GATE 2017 EE – Question 37

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 · Multiple choice

Consider the differential equation $(t^2 - 81)\dfrac{dy}{dt} + 5ty = \sin(t)$ with $y(1) = 2\pi$. There exists a unique solution for this differential equation when $t$ belongs to the interval

$(-2, 2)$
$(-10, 10)$
$(-10, 2)$
$(0, 10)$

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Correct answer: (A) $(-2, 2)$

Explanation

Dividing by $t^2 - 81$ gives a linear equation whose coefficients are continuous except at $t = \pm 9$. A unique solution exists on the largest interval around the initial point $t = 1$ that avoids those two points, which is $(-9, 9)$. Of the options, only $(-2, 2)$ lies within it. The other intervals contain $\pm 9$.