Discussion
Let \(\phi_{1}, \phi_{2}\) be linearly independent solutions of the linear differential equation :
\[ y'' + a_{1}y' + a_{2}y = 0, \]
where \(a_{1}, a_{2}\) are constants, then the Wronskian \(W(\phi_{1}, \phi_{2})\) is constant if and only if :
Answer: A
The Wronskian is constant if and only if \(a_{1} = 0\).
By Abel's formula, the Wronskian of two solutions of \(y'' + a_{1}y' + a_{2}y = 0\) satisfies \(W' = -a_{1}W\), so
\[ W(x) = W(x_{0})\, e^{-a_{1}(x - x_{0})} \]
Because \(\phi_{1}, \phi_{2}\) are linearly independent, \(W(x_{0}) \ne 0\). The exponential is constant exactly when \(a_{1} = 0\).
The coefficient \(a_{2}\) does not enter Abel's formula at all, so conditions on \(a_{2}\) are irrelevant; and \(a_{1} \ne 0\) makes the Wronskian grow or decay exponentially, not stay constant.
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