How do you solve e ^ { 2x } = 3x ^ { 2}e2x=3x2?
2 Answers
Explanation:
This has no solution in terms of elementary functions, but is expressible in terms of the Lambert W function...
Suppose:
(-x)e^(-x) = +-1/sqrt(3)(−x)e−x=±1√3
Squaring both sides, we find:
x^2e^(-2x) = 1/3x2e−2x=13
Multiplying both sides by
3x^2=e^(2x)3x2=e2x
The Lambert W function (actually a multi-valued function or family of functions) satisfies:
ze^z = a" " <=> " " z = W_n(a)zez=a ⇔ z=Wn(a) for some integernn
The two real valued branches of
For our example, we have:
-x = W_n(+-1/sqrt(3))−x=Wn(±1√3)
That is:
x = -W_n(+-1/sqrt(3))x=−Wn(±1√3)
The real valued solution is given by:
x = -W_0(1/sqrt(3)) ~~ -0.39x=−W0(1√3)≈−0.39
Here are the graphs of the functions
graph{(y-e^(2x))(y-3x^2) = 0 [-2.657, 2.343, -0.34, 2.16]}
See below.
Explanation:
We have
Regarding the Lambert function
https://en.wikipedia.org/wiki/Lambert_W_function
we have
with real value for