Project ForgeEngineering Mathematics
Calculus
GATE 2023
Medium
The initial value problem dy/dt+2y=0, y(0)=1, is solved numerically using forward Euler's method with a constant positive time step Δt. Let yn represent the numerical solution after n steps. The condition |y(n+1)|≤|yn| is satisfied if and only if Δt does not exceed ______.
1(± 0.05 accepted)
Explanation
Forward Euler gives y(n+1)=yn+Δt(−2yn)=yn(1−2Δt). The condition |y(n+1)|≤|yn| requires |1−2Δt|≤1, i.e. 0≤2Δt≤2, so 0≤Δt≤1. Since Δt>0, the threshold is Δt≤1.
Source: GATE ME 2023 Official Paper (2023)
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