Part CCSIR NET June 2025the-coefficients-have-period-2pi-so-a-shifted-solution-is-still-in-the-two-dimensional-span

The coefficients have period 2pi so a shifted solution is still in the two dimensional span

Consider the ordinary differential equation (ODE $) d^{2} y /$ dx $^{2} + cos (x)$ dy/dx + sin(x) y = 0. Let $φ_{1} (x), φ_{2} (x)$ be solutions of the ODE, satisfying $φ_{1} (0) = 1, d φ_{1} /$ dx(0) = 0, and $φ_{2} (0) = 0, d φ_{2} /$ dx(0) = 1. Then which of the following statements are true?

A. $φ_{1} (x + 2 π)$ is also a solution of ODE✓
B. $φ_{2} (x + 4 π)$ is also a solution of ODE✓
C.There are NO constants a, b such that $φ_{2} (x + 4 π) = a φ_{1} (x) + b φ_{2} (x)$
D.There exist $a, b \in R$ such that $φ_{1} (x + 2 π) = a φ_{1} (x) + b φ_{2} (x)$ ✓

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 standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

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

The coefficients have period 2pi so a shifted solution is still in the two dimensional span

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