tan(x) = A exp(2x) solvable?

Question

This equation originates from an attempt to inscribe a circle
into a leaf-shaped area formed by two logarithimic spirals.

Answer 1

In explicit terms, no. Numerical approximations can be done, solutions do exist, the first real root > 0 being less than π/2. Another way to do it is to use series approximations of both functions, and solve, as in:

x + 1/3 x³ = a(1 + 2x + 2x²)

That does have an explicit real root, which is as follows (hold on!):

x = 2a
+ (2^1/3 (9-18a-36a²) / 9(3a-12a²-16a³ + √(4-24a+9a²+88a³+48a^4))^1/3
- (1/(2^1/3)) (3a-12a²-16a³ + √(4-24a+9a²+88a³+48a^4))^1/3