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 $\varphi_{1}(x), \varphi_{2}(x)$ be solutions of the ODE, satisfying $\varphi_{1}(0) = 1, d\varphi_{1}/$ dx(0) = 0, and $\varphi_{2}(0) = 0, d\varphi_{2}/$ dx(0) = 1. Then which of the following statements are true?

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