Example 2.3.1.
Let us consider the equation
\begin{equation}
x'' + 6 x' + 5x = \sin 2t.\tag{2.3.1}
\end{equation}
The solution to the corresponding homogeneous equation, \(x'' + 6 x' + 5x = 0\text{,}\) is
\begin{equation*}
x_h = c_1 e^{-5t} + c_2 e^{-t}.
\end{equation*}
To find a particular solution, we can use the method of undetermined coefficients and assume that the solution has the form
\begin{equation*}
x_p = A \cos 2t + B \sin 2 t.
\end{equation*}
If we carry out the appropriate calculations, we will obtain a particular solution
\begin{equation*}
x_p = - \frac{12}{145} \cos 2t + \frac{1}{145} \sin 2t.
\end{equation*}
Thus, the general solution is
\begin{equation*}
x = x_h + x_p = c_1 e^{-5t} + c_2 e^{-t} - \frac{12}{145} \cos 2t + \frac{1}{145} \sin 2t.
\end{equation*}
Notice that all solutions of (2.3.1) will approach the particular solution as \(t \to \infty\text{.}\)