How to solve this equation for x (May require product log or Lambert-W function)

Question

I need help solving the following for t (by hand):

$$ A=Bt+Ce^{Dt} $$ where A, B, C, and D are constants.

Is this possible? I'm guessing the Lambert-W function is required.

Thank you

Answer 1

If $A-Bt = s$, so $t = A/B - s/B$, the equation becomes $$ s = C \exp(AD/B - Ds/B)$$ i.e. $$ s \exp(Ds/B) = C \exp(AD/B) $$ and then with $u = Ds/B$, $$ u \exp(u) = \dfrac{CD}{B} \exp(AD/B)$$ So we have $$u = W\left( \dfrac{CD}{B} \exp\left(\frac{AD}{B}\right)\right)$$ i.e. $$ t = \dfrac{A}{B} - \dfrac{1}{D} W\left(\dfrac{CD}{B} \exp\left(\frac{AD}{B}\right)\right)$$ where $W$ is any branch of the Lambert W function.