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Part CCSIR NET December 2024both-roots-negative-leaves-nothing-to-grow

Both roots negative leaves nothing to grow

Consider the non-homogeneous ordinary differential equation (ODE)d2y/) d^{2}y/dx2+5^{2} + 5dy/dx + 6y = sin(e^(−5x)), x > 0. Then which of the following statements are true?

  1. A.Every solution of the ODE is bounded on (0,∞)(0, \infty)✓ Correct
  2. B.There exists a solution of the ODE which is unbounded on (0,∞)(0, \infty)
  3. C.Every solution of the ODE is unbounded on (0,∞)(0, \infty)
  4. D.Every solution of the ODE tends to zero as x→∞x \to \infty✓ 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 boundary and endpoint.

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50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

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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