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The final particular solution is $X_p(t) = 2te^{-t}$. This is obtained by substituting the solved coefficient $A=2$ back into the resonant trial solution $x=Ate^{-t}$.

Conditions: The ODE is $\ddot{x}+3\dot{x}+2x=2e^{-t}$.; The coefficient $A$ has been determined to be $2$.; The trial solution form is $x=Ate^{-t}$.

The final particular solution is $X_p(t) = 2te^{-t}$. This is obtained by substituting the solved coefficient $A=2$ back into the resonant trial solution $x=Ate^{-t}$.

Conditions: The ODE is $\ddot{x}+3\dot{x}+2x=2e^{-t}$.; The coefficient $A$ has been determined to be $2$.; The trial solution form is $x=Ate^{-t}$.