I have been in trouble finding a general solution for the following differential equation. \begin{equation} \ddot{y} + a_{1}\dot{y} + (a_{2}+a_{3}e^{-2a_{4}t})y = a_{5}e^{-a_{4}t} \end{equation} with the following initial conditions: \begin{equation} y(0) = \dot{y}(0)=0 \end{equation} I have tried a few methods to solve this equation but none of them seems promising.

  1. Variation of parameters: I need to know at least one particular solution to apply this method but I could not find one. Is there any method to find one? (Reference: Chapter$$ of https://www.jirka.org/diffyqs/html/nonhomogsys_section.html)

  2. Laplace transform: Let Y be the Laplace transform of y, then the Laplace transform of this equation is: \begin{equation} (s^{2}+a_{1}s+a_{2})Y(s) + a_{3}Y(s+2a_{4}) = \frac{a_{5}}{s+a_{4}} \end{equation} but I have no idea how to deal with $Y(s+2a_{4})$ term. Moreover, comparison of the first term and the RHS implies that $Y(s)$ has an order of $s^{-3}$ but comparison of the second term and the RHS implies that $Y(s+2a_{4})$ has an order of $s^{-1}$, which seems to be impossible...

I would appreciate if you could give me any piece of advice. Thank you.


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