= Solution
The second component is resonant. Differentiating $t e^{\lambda_2t}$ supplies the term missing from the exponential ansatz:
$$
(\partial_t-A)(t\mathbf a_2e^{\lambda_2t})=\mathbf a_2e^{\lambda_2t}.
$$
Thus a <particular integral> is
$$
\boxed{\mathbf x_p=\frac{\mathbf a_1e^{i\omega_1t}}{i\omega_1-\lambda_1}
+t\mathbf a_2e^{\lambda_2t}.}
$$
The linearly growing oscillatory envelope is forced <resonance in a differential equation>; it does not contradict bounded oscillation of the <complementary function>. The worked initial-value solution in the question's root Solution uses this same resonant mechanism.
Back to article page
