Part CCSIR NET December 2024the-sign-of-the-larger-root-decides-boundedness

The sign of the larger root decides boundedness

Consider the ordinary differential equation (ODE $) d^{2} y /$ dx $^{2} + 2 (e^{(} 2 x) - 1) ($ dy/dx $) + α e^{(} 4 x) y = 0, x \in R$ , where $α \in R$ . Note that the ODE transforms into an equation with constant coefficients under the change of independent variable given by t = e^(2x)/2. Then which of the following statements are true?

A.For $α = 1$ , all the solutions of the ODE tend to zero as $x \to \infty$ ✓
B.For $α = 0$ , there exists a solution of the ODE which tends to 1 as $x \to \infty$ ✓
C.For $α = - 1$ , there exists an unbounded solution of the ODE on $R$ ✓
D.For $α = 2$ , there exists an unbounded solution of the ODE on $R$

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

The sign of the larger root decides boundedness

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