How do we solve the equation for this exponential function?

This is

Maybe it's just my lack of understanding, but
I don't expect Sympy to answer such a complicated formula.

If you want to know how to answer, it is a little different from the purpose of Stackoverflow, but
I'll give you a little hint.

You can solve it by multiplying both sides with Ln (natural log) and expanding the formula a couple of times.

 exp(-1/x**2) = exp(-2/x**2) - exp(-3/x**2)

Put 1/x**2 as y and

exp(-y)=exp(-2y)-exp(-3y)

After a slight deformation,

exp(y)+exp(-y)=1

If you draw a graph, you can see that there is no solution.
But it's a real world story, and if you want to bring in an imaginary number, go further.

If y=zi is placed,

exp(zi)+exp(-zi)=1

Using Euler's formula,

 cos(z)+isin(z)+cos(-z)+isin(-z)=1

When asked,

z=± //3+2 nn (where n is an integer)

Go back to where you were,

y=zi
  = (± //3+2 nn)i

1/x**2=(± //3+2 nn)i

Therefore,

x=±sqrt(1/(± //3+2 nn)i)

I'm not confident, but...

Sympy Live looks like this↓

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