Differential Equations Mastery Hub: The Industry Foundation

In the context of solving linear ODEs using the method of undetermined coefficients, what is the primary rationale for modifying the assumed particular solution when the forcing function shares terms with the complementary solution?

Consider a system of linear ODEs represented by $\mathbf{x}'(t) = A\mathbf{x}(t)$, where $A$ is a $2 \times 2$ matrix with distinct real eigenvalues $\lambda_1$ and $\lambda_2$. If $\lambda_1 > 0$ and $\lambda_2 < 0$, what is the long-term behavior of solutions originating from a generic initial condition $\mathbf{x}(0)$ (not an eigenvector corresponding to $\lambda_2$)?

For a second-order linear ODE $ay'' + by' + cy = f(t)$, where $a, b, c$ are constants and $f(t)$ is continuous, what is the fundamental implication of the Wronskian of two solutions $y_1(t)$ and $y_2(t)$ being non-zero on an interval?