Discussion
A particular integral of \( y'' - y = e^{x} \) requires a trial solution of the form
Answer: D
The auxiliary equation \( m^{2} - 1 = 0 \) has \( m = 1 \) as a root, so \( e^{x} \) already satisfies the homogeneous equation and \( Ae^{x} \) can never produce a non-zero right-hand side. Multiplying by \( x \) breaks the degeneracy; a factor \( x^{2} \) would be needed only for a *repeated* root
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