Solving for t in Lambert W Function: a=bt+e^(ct)

[fisico]

How can I use the W function to solve for t?

a = bt + e^(ct),

where a,b,c,e are known constants. e is Euler's number. t is the unknown?

I have only seen examples where a = 0. Thanks.

 

[HallsofIvy]

Do a little thinking and a little algebra to get it into the right form.
Lambert's W function is the inverse function to f(x)= xe^x so you have to change variables to get your equation into that form. From
a= bt+ e^ct, a- bt= e^ct. Okay, let u= a- bt. Then t= -(u-a)/b so
$$e^{ct}= e^{\frac{-cu}{b}}e^{\frac{a}{b}}= u$$
so
$$e^{\frac{a}{b}}= ue^{\frac{cu}{b}}$$
Now let $x= \frac{cu}{b}$ so $u= \frac{b}{c}x$ and
$$e^{\frac{a}{b}}= \frac{b}{c}xe^x$$
Finally,
$$\frac{c}{b}e^{\frac{a}{b}}= xe^x$$
so that
$$x= W(\frac{c}{b}e^{\rac{a}{b}})$$

 

[fisico]

Thanks a lot. That's cool!