Solving an equation of the type $axe^{qx} + be^{rx} + cx + d = 0$

Question

I need to solve an equation of the type, $axe^{qx} + be^{rx} + cx + d = 0$

I tried but couldn't solve it.

Does anyone have an idea how to solve this(for x)?

Thanks

Answer 1

The equation as stated does not admit a symbolic (closed-form) solution. The following variations of it can be solved analytically:

$$ \begin{equation} \begin{split} axe^{qx}+cx & = 0 \quad \text{(Solution: } \frac{1}{q}\log(-\frac{c}{a}) \text{)} \\ axe^{qx}+d & = 0 \quad \text{(Solution: } \frac{1}{q}W(-\frac{dq}{a}) \text{, where W is the Lambert W function)} \\ \end{split} \end{equation} $$

More general forms such as the one you provided are out of reach from an analytical standpoint, but may easily be tackled numerically with any variation of Newton's method, especially since the derivatives can be computed analytically and therefore rapidly evaluated to high precision. Additionally, the optimization function is convex for many parameter choices, so you can expect rapid convergence to a zero without problems.