Discussion
A particular integral of \( y'' - y = e^{2x} \) is
Answer: A
Substituting \( y = Ae^{2x} \) replaces each derivative by a power of two:
\[ 4A e^{2x} - Ae^{2x} \;=\; e^{2x} \quad \Longrightarrow \quad 3A \;=\; 1 \quad \Longrightarrow \quad A \;=\; \tfrac{1}{3}. \]
Equivalently \( 1/(m^{2}-1) \) evaluated at \( m = 2 \) gives \( 1/3 \). A factor of \( x \) would be needed only if two were a root of the auxiliary equation
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