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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?

  1. A.lim⁡(t→∞)t\lim(t\to\infty) t⁻2x(t)y(t)=0^{2}x(t)y(t) = 0
  2. B.x(1) = 0, y(1/2) = 0
  3. C.x(1/2) = 0, y(1) = 0✓ Correct
  4. D.lim⁡(t→∞)t\lim(t\to\infty) t⁻2x(t)y(t)=2^{2}x(t)y(t) = 2✓ Correct

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, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

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

More on this topic

The chapter behind this: Linear ODE, Wronskian and systems — free to read

From Ordinary Differential Equations › Linear ODE, Wronskian, variation of parameters, systems

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