Part CCSIR NET December 2024a-substitution-removes-both-exponentials

A substitution removes both exponentials

If x = x(t), y = y(t) is the solution of the initial value problem dx/dt = x − 4e^(−2t)y, dy/dt = e^(2t)x − y, x(0) = 1, y(0) = 1, then which of the following statements are true?

A. $lim (t \to \infty) t$ ⁻ $^{2} x (t) y (t) = 0$
B.x(1) = 0, y(1/2) = 0
C.x(1/2) = 0, y(1) = 0✓
D. $lim (t \to \infty) t$ ⁻ $^{2} x (t) y (t) = 2$ ✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: W(f, g) ≡ 0 implies f and g are linearly dependent

A substitution removes both exponentials

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