GATE 2023 MA – Question 49

Ordinary Differential Equations · Higher Order Equations and Systems · 2 marks · Multiple select

For x′=y,y′=x with x(0)=α,y(0)=β, which choices give |x(t)|+|y(t)|→0 as t→∞?

α=1,β=−1
α=1,β=1
α=1.01,β=−1
α=1,β=1.01

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Show answer and explanation

Correct answer: (A) α=1,β=−1

Explanation

Let u=x+y and v=x−y. Then u′=u and v′=−v, so u=(α+β)eᵗ and v=(α−β)e⁻ᵗ. Decay occurs precisely when α+β=0, which only A satisfies. Small positive deviations still excite the growing mode. **A**.

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