Discussion
Consider the following statements for any solution \(\phi\) of the differential equation with constant coefficient :
\[ y'' + a_{1}y' + a_{2}y = 0, \]
where \(a_{1}, a_{2}\) are constants.
(I) \(\phi \to 0\) as \(x \to \infty\), if the real parts of the roots of the characteristic polynomial are negative.
(II) \(\phi \to 0\) as \(x \to \infty\), if the real parts of the roots of the characteristic polynomial are positive.
Then :
Answer: A
Only I is true.
Every solution is a combination of terms \(e^{rx}\), \(x e^{rx}\) or \(e^{\alpha x}\cos\beta x\), \(e^{\alpha x}\sin\beta x\), where \(r\) or \(\alpha \pm i\beta\) are the characteristic roots. If every root has negative real part, each term carries a decaying exponential and \(\phi \to 0\).
If the real parts are positive, the terms grow: for \(y'' - 3y' + 2y = 0\) the roots are 1 and 2 and \(\phi = e^{x}\) tends to infinity. So II is false (only the zero solution would tend to 0).
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