I'm trying to solve equations like the following one:
$$5 + 3x – 4x^3 = e^{x^2}$$
I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W function.
I've managed to bring the equation to:
$$e^{-x^2}(5 + 3x – 4x^3) = 1$$
But I can't bring the exponent and the second term on the LHS to the same value, in order to apply Lambert W function.
Also I tried to search on the Internet, but I never found a equation from a simular type, all of them were from the type:
$$5^x = 7x$$
This type of equation is fairly easy to solve using Lambert W function, but it doesn't help me solving an equation from the first type.
Also I couldn't come up with another idea how to get solution, except for Lambert W function. Can you please help me?
Best Answer
There are a few problems with addition.
$$W(x+a)=?$$
$$\log(x+a)=?$$
The inverse functions of exponential related functions don't like addition on the inside. Addition on the outside is perfectly fine though.
$$\log(x)+a=\log(e^ax)$$
Except even then, the Lambert W Function won't simplify if you have addition on the outside... most of the time. And this question isn't one of the ones that simplifies.
So the best is to go with linear approximation methods.