Solving symbolically a trascendental equation containing an exponential

Question

Is it possible to solve symbolically this equation for $x$: $$\exp \left(-x^2\right)=\frac{c_1}{\sqrt{c_2-c_3 x}}$$

Exp[-x^2] == c1/Sqrt[c2 - c3 x]

$c_1$, $c_2$ and $c_3$ are positive constants, with $\frac{c_1}{\sqrt{c_2}}<1$. I am interested in the negative solution, $x<0$. A typical plot of the two functions at the RHS and LHS of the equation is

I tried an expansion at zero to the second order of the two sides of the equation, but the result besides being complicated does not seem to give a good approximation of the result. On the other side, it is easy to find a numerical solution to the problem:

NSolve[Exp[-x^2] == c1/Sqrt[c2 - c3 x] && x < 0, x]

Thanks in advance.

Answer 1

It seems you have a Transcendental equation, so you will need to approximate, and that implies choices:

• Wich kind of approximation
• To which order
• Around what value?

In this example PadeApproximant, order {1,2}, around zero.

Block[
    {
        eqn = (Exp[-x^2]==c1/Sqrt[c2-c3 x]),    (* Define equality *)
        lhs, rhs, pa
    },
    {lhs,rhs}=List@@ApplySides[Power[#,2]&,eqn];(* Manipulate as necesary *)
    pa=PadeApproximant[lhs, {x, 0, {1,2}}];     (* Algebraic aprox*)
    x/.FullSimplify@Solve[pa == rhs, x]         (* Solve *)
]

How good was the PadeApproximant? I approximated around zero, you may want to try something else.

If you are willing to tolerate a messier expression, a better approximation could be around (c2- 4*c1^2)/c3, ie Solve[rhs==1/4,x]

PadeApproximant[lhs, {x, (c2- 4*c1^2)/c3, {1,2}}]