Question29 Consider the functions [math] and [math]. Use the Wronskian to decide whether these functions are linearly independent on the interval [math]. Which of the following gives the correct answer and appropriate justification for it? Select one alternative: [math] has nonzero determinant for every value of [math], so the functions are linearly independent. [math]has determinant equal to zero for every value of [math], so the functions are linearly independent. [math]has determinant equal to zero for every value of [math], so the functions are linearly dependent. [math]has nonzero determinant for every value of [math], so the functions are linearly dependent. ResetMaximum marks: 1 Flag question undefined

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BlackTom AI · Accepted Answer

We start by restating the problem and the available choices to ensure clarity about what is being evaluated.
Question: Use the Wronskian to decide whether the given pair of functions are linearly independent on the interval [ ] and identify which option provides the correct justification.
Answer options:
1) [e2t3e2t2e2t6e2t] has nonzero determinant for every value of t, so the functions are linearly independent.
2) [e2t3e2t2e2t6e2t] has determinant equal to zero for every value of t, so the functions are linearly independent.
3) [e2t3e2t2e2t6e2t] has determinant equal to zero for every value of t, so the functions are linearly dependent.
4) [e2t3e2t2e2t6e2t] has nonzero determinant for every value of t, so the functions are linearly dependent.

Option-by-option analysis:
Option 1: Nonzero determinant for every value of t implies the Wronskian never vanishes on the interval. In many texts this is presented as a strong sufficient condition for linear independence: if W(y1, y2)(t) ≠ 0 for all t in the interval, then the two functions are linearly independent on that interval. This aligns with the intuitive idea that the functions and their derivatives form a non-singular 2×2 matrix at every t, preventing any constant c1, c2 (not both zero) from satisfying c1 y1 + c2 y2 = 0 identically. A potential subtlety to keep in mind is that some formulations emphasize that a nonzero Wronskian at a single point already guarantees independence for solutions of linear ODEs, but a nonzero Wronskian everywhere is certainly consistent with independence and provides a robust justification.
Option 2: determinant equal to zero for every t, and independence. This is inconsistent with the standard criterion: if the Wronskian is identically zero on an interval, the functions may be linearly dependent; however, it is not correct to conclude independence from a Wronskian that is identically zero. In fact, identically zero Wronskian is a hallmark of potential dependence, so stating independence here is incorrect and stems from misapplying the criterion.
Option 3: determinant equal to zero for every t, and dependence. This matches the classical implication: if W(y1, y2)(t) ≡ 0 on the interval, the functions are often linearly dependent (at least for many standard classes of functions, and under mild regularity conditions). This option correctly identifies dependence as the consequence of a vanishing Wronskian across the interval, making it a plausible and commonly taught justification. The caveat to note is that a vanishing Wronskian does not universally guarantee dependence for arbitrary non-solution functions, but in the typical course context it is presented as evidence of dependence.
Option 4: nonzero determinant for every t, and dependence. This contradicts the fundamental link between a nonzero Wronskian and independence: a consistently nonzero Wronskian should indicate independence, not dependence. Thus this option is logically inconsistent with standard theory and can be dismissed as incorrect.

Synthesis and common misconceptions:
- A nonzero Wronskian at all t strongly supports independence; a Wronskian that is identically zero supports dependence in typical linear ODE contexts. The subtlety that a nonzero Wronskian at a single point suffices for independence in many treatments is related to the fact that the Wronskian either is identically zero or never zero on an interval for solutions of homogeneous linear ODEs with continuous coefficients, but the question here asks for the given options' justification, not to restate the full theorem.
- It is important to distinguish between a Wronskian that vanishes for all t (which suggests dependence) and one that does not vanish (which suggests independence). The misstep is to claim independence from a Wronskian that identically vanishes, or to claim dependence from a nonzero Wronskian.

In summary, while Option 2 incorrectly states independence and Option 4 incorrectly asserts dependence, Option 3 correctly links an identically zero Wronskian to dependence, and Option 1 correctly links a Wronskian that is nonzero for all t to independence. The evaluation hinges on whether the given phrasing is interpreted strictly as a universal nonvanishing condition (Option 1) or as a standard identically zero condition implying dependence (Option 3), with Option 3 being the classic correct interpretation in many standard treatments, and Option 1 representing a valid but stronger sufficient condition. The key ideas are the behavior of the Wronskian and its implications for linear independence or dependence on the interval.

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