a resonant forcing term
Push a swing at a random rhythm and not much happens; push it at exactly its natural rhythm and the arcs grow and grow. A resonant forcing term is the analogue in these equations: a piece of the forcing whose shape coincides with a natural mode of the system — that is, with a complementary solution. Forcing a system on a frequency it already 'likes' produces an outsized, growing response.
Algebraically, a forcing term is resonant when it is (a constant multiple of) one of the homogeneous solutions in y_c. For a constant-coefficient equation that means the exponent or frequency in g matches a root of the characteristic equation: forcing e^(kx) when k is a characteristic root, or forcing cos(w0 t) / sin(w0 t) when w0 is the natural frequency (i.e. the homogeneous solutions are cos(w0 t), sin(w0 t)). When that happens, the ordinary trial solution coincides with a complementary solution and fails, which is exactly the situation the modification (multiply-by-x) rule was built for. The particular solution then carries an extra factor of x (or x^2 for a repeated root) and grows in time — these growing pieces are sometimes called secular terms.
Resonance is the physically vivid case where forcing and natural behaviour line up: bridges, buildings, and circuits can build dangerously large amplitudes this way. One honest correction to the cartoon, though: a forcing term produces UNBOUNDED growth only in the idealized undamped case. With any real damping, the homogeneous frequency shifts slightly and a sinusoidal drive at the resonant frequency gives a large but FINITE steady-state amplitude, not an x-growing solution. The clean x-times-oscillation 'pure resonance' is the undamped limit.
In y'' + y = cos(t), the forcing cos(t) coincides with the homogeneous solution cos(t) (natural frequency 1), so it is resonant. The trial A cos t + B sin t fails; the x-rule gives y_p = (t/2) sin(t), an oscillation whose amplitude grows linearly in t.
A resonant term matches a natural mode, triggering the x-rule and a growing response.
Unbounded x-growth is the undamped idealization; real damping turns resonance into a large but finite steady-state peak, so 'resonance requires zero damping' is a myth — light damping still resonates strongly.