For a constant-coefficient equation

$$ay''+by'+cy=g(t),$$

the solution can be written as

$$y=y_h+y_p,$$

where $y_h$ solves the homogeneous equation and $y_p$ is one particular response to the forcing $g(t)$.

When $g(t)$ is built from exponentials, polynomials, sines or cosines, the method of undetermined coefficients guesses a particular solution of a matching form and determines its unknown coefficients by substitution.

If the guessed form overlaps a homogeneous solution, it must be multiplied by a sufficient power of $t$ to obtain an independent trial function.

This overlap is the mathematical source of resonant terms such as

$$t\sin(\omega t).$$

The method is efficient for structured forcing but is not a general solution technique for arbitrary $g(t)$.