Discussion
If \(\phi_{1}\) and \(\phi_{2}\) are two linearly independent solutions of \(y'' + ay = 0\), where \(a\) is a constant, on an interval I. Then which of the following statements about the Wronskin \(W(\phi_{1} \phi_{2})(x)\) is true ?
Answer: A
The Wronskian is a constant function.
For \(y'' + p(x)y' + q(x)y = 0\), Abel's formula gives \(W(x) = W(x_{0})\exp\left(-\int_{x_{0}}^{x} p(t)\, dt\right)\). Here there is no \(y'\) term, so \(p = 0\) and \(W\) is constant. Directly: \(W' = \phi_{1}\phi_{2}'' - \phi_{1}''\phi_{2}\), which is \(\phi_{1}(-a\phi_{2}) - (-a\phi_{1})\phi_{2} = 0\).
Because \(\phi_{1}, \phi_{2}\) are linearly independent, the constant is non-zero, so D (identically zero) is false, and a non-zero constant is neither increasing nor decreasing.
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