Exercise · ZW-WRONSKIAN-009 · ★★★ Advanced
Let \(y_1,y_2\) be solutions of \(y''+py'+qy=0\) with \(p,q\) continuous, and \(W(y_1,y_2)(t_0)=0\) at some \(t_0\). Which is true?
Choose one option.
A They are independent, because \(W\) vanishes only at \(t_0\) B \(W\) changes sign at \(t_0\) C \(W\) vanishes for every \(t\) and the solutions are linearly dependent D Nothing can be concluded without knowing \(p\) and \(q\) Show answer C. \(W\) vanishes for every \(t\) and the solutions are linearly dependent← Back to the question
Explanation By Abel's formula \(W(t)=W(t_0)e^{-\int_{t_0}^{t} p}\): if it is zero at one point, it is zero everywhere. For solutions of the same equation there are no isolated zeros .
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