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Part CCSIR NET December 2025sin-of-1-over-x-is-approximately-1-over-x-for-large-x-so-r-grows-like-x-squared-not-a-nuisance-oscillation

Sin of 1 over x is approximately 1 over x for large x so r grows like x squared not a nuisance oscillation

Consider the ordinary differential equation (ODE) y″ + r(x)y = 0, where r(x)=x3sin⁡(1/x)r(x) = x^{3}\sin(1/x) for x≠0, and r(0)=0. Then which of the following statements are true?

  1. A.There exists a non-trivial solution φ\varphi of ODE, and α,β∈R,α<β\alpha,\beta\in\mathbb{R}, \alpha<\beta such that φ\varphi has infinitely many zeros in [α,β][\alpha,\beta].
  2. B.There exists a non-trivial solution φ\varphi of ODE, and a sequence {xnx_{n}} such that xn→∞x_{n}\to\infty and φ(xn)=0\varphi(x_{n})=0 for all n∈Nn\in\mathbb{N}.✓
  3. C.If φ1,φ2\varphi_{1},\varphi_{2} are two linearly independent solutions of ODE, then their Wronskian is a constant function on R\mathbb{R}.✓
  4. D.There exists a non-trivial solution of ODE which vanishes at most at one point in (0,∞)(0,\infty).

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: Every boundary value problem has a Green's function

More on this topic

The chapter behind this: Sturm–Liouville problems and Green's functions — free to read

From Ordinary Differential Equations › Sturm–Liouville problems and Green's functions

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