Part CCSIR NET June 2025with-no-first-derivative-term-the-wronskian-must-be-constant

With no first derivative term the wronskian must be constant

For a continuous function q defined on $R$ , consider the ordinary differential equation (ODE $) d^{2} y /$ dx $^{2} + q (x) y = 0, x \in R$ . Then which of the following statements are FALSE?

A.There exists a q such that cos(x) and e^x cos(x) are solutions of ODE✓
B.There exists a q such that sin(x) and cos(x) are solutions of ODE
C.There exists a q such that e^x sin(x) and e^x cos(2x) are solutions of ODE✓
D.There exists a q such that xe^x and x(x − 1)e^x are solutions of ODE✓

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

With no first derivative term the wronskian must be constant

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