GATE 2024 PI – Question 46
$x^{\prime\prime}(t)+4x(t)=e^{-t}$ for t ≥ 0, with x(0) = x′(0) = 0. Find x(π/8) to three decimals.
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Correct answer: 0.064 to 0.065
Explanation
A particular solution is $e^{-t}/5$; add $A\cos2t+B\sin2t$. Initial conditions give A = −1/5 and B = 1/10. Therefore $x(t)=e^{-t}/5-\cos2t/5+\sin2t/10$. At t = π/8 this is about 0.06434, rounding to **0.064**.