Discussion
Let \(A\) be a \(2 \times 2\) non-diagonalizable real matrix with a real eigenvalue \(\lambda\) and \(v\) be an eigenvector of \(A\) corresponding to \(\lambda\). Which one of the following is the general solution of the system \(y' = Ay\) of first-order linear differential equations?
Answer: C
The general solution is \(c_{1}e^{\lambda t}v + c_{2}e^{\lambda t}(tv + u)\), with \((A - \lambda I_{2})u = v\).
A non-diagonalisable \(2 \times 2\) matrix has a single eigenvector direction \(v\), and a generalised eigenvector \(u\) with \((A - \lambda I)u = v\). Check the second solution \(y = e^{\lambda t}(tv + u)\):
\[ \begin{aligned} y' &= \lambda e^{\lambda t}(tv + u) + e^{\lambda t}v \\ Ay &= e^{\lambda t}(t\lambda v + \lambda u + v) \end{aligned} \]
These agree. \(te^{\lambda t}v\) alone is not a solution (its derivative has an extra \(e^{\lambda t}v\) that \(A\) cannot produce), which rules out the first option; \(t^{2}e^{\lambda t}v\) fails the same way; and \(te^{\lambda t}(v + u)\) misplaces the factor \(t\), so it does not satisfy the system either.
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