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Part BCSIR NET December 2025no-first-derivative-term-means-the-wronskian-is-constant-everywhere-not-just-computable-at-0

No first derivative term means the wronskian is constant everywhere not just computable at 0

For aRa \in \mathbb{R}, let y1(x)y_{1}(x) and y2(x)y_{2}(x) be solutions of the differential equation y+(e(x2)+cosx)y=0y'' + (e^(x^{2}) + \cos x)y = 0 such that y1(0)=3,y1(0)=1,y2(0)=5,y2(0)=ay_{1}(0) = 3, y_{1}'(0) = -1, y_{2}(0) = -5, y_{2}'(0) = a. Suppose W(y1,y2)(x)W(y_{1},y_{2})(x) denotes the Wronskian of y1y_{1} and y2y_{2}. If W(y1,y2)(1/2)=4W(y_{1},y_{2})(1/2) = 4, then the value of a is

  1. A.2
  2. B.3
  3. C.4
  4. D.−3

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

More on this topic

The chapter behind this: Linear ODE, Wronskian and systems — free to read

From Ordinary Differential EquationsLinear ODE, Wronskian, variation of parameters, systems

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