Solution for Transcendental Equation $B+x\sin{\left(\frac{A}{x}\right)}=0$

Question

I am trying to solve a transcendental equation of the form:

$$B+x\sin{\left(\frac{A}{x}\right)}=0,$$ where both $A$ and $B$ are constants.

What would be the best approach to solve it?

Answer 1

I do not think that any closed form solution exists for this class of equations even if we rewrite them as C y = Sin[y] using y = A / x and C = - B / A.
Numerical techniques should be used and the simplest would be Newton second order method
$$x_{new} = x_{old} - f[x_{old}] / f'[x_{old}]$$
The problem is that there could be many roots (or no root at all) depending on the respective values of A and B.
To illustrate with an example, let me consider the case where A = e and B = Pi / 10. A plot of the function shows that the largest root is close to x = 0.75(the exact solution is x = 0.762183). Then, let us start the Newton procedure at $$x_{old} = 1$$ Then, the successive iterates will be 0.749080, 0.762427, 0.762183.