Solving $B a^t + C a^{2t} + D t a^t + E t^2 + F t + G = 0$

Question

I'm working my way throught a complex system of equations, and I reached a point where what I need to solve for $t$ may be written as follows (everything else is constant): $$B a^t + C a^{2t} + D t a^t + E t^2 + F t + G = 0$$ I suppose I should somehow write it in the form $a^x = bx + c$ and solve it with the Lambert W function. But how do I reach this stage?

Answer 1

$$Ba^t+Ca^{2t}+Dta^t+Et^2+Ft+G=0$$ $$Be^{a\ln(t)}+C\left(e^{a\ln(t)}\right)^2+Dte^{a\ln(t)}+Et^2+Ft+G=0$$ The equation contains more than one power of $e^{a\ln(t)}$ and more than one power of $t$. Therefore it's unlikely that the equation can be solved in terms of Lambert W or Generalized Lambert W.

For solving in terms of Lambert W, we need equations in the form like $$f(t)e^{f(t)}=c$$ or $$R_1(t)e^{R_2(t)}=c,$$ where $c$ is a constant, and $R_1(t),R_2(t)$ are non-constant rational expressions of $t$.

For solving in terms of Generalized Lambert W of Mező et. al., we need equations of the form $$R(t)e^{c_0+c_1t}=c,$$ where $c$ is a constant, and $R(t)$ is a non-constant rational expression of $t$.

I don't now if your general equation is solvable in terms of Generalized Lambert W of Castle.

[Mező 2017] Mező, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553

[Mező/Baricz 2017] Mező, I.; Baricz, Á.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)

[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018