Klein-Gordon equation for a charged particle in space varying electromagnetic fields-A systematic study via Laplace transform

Exact solutions of the Klein-Gordon equation for a charged particle in the presence of three spatially varying electromagnetic fields, namely, (i) $\vec{E}=\alpha\beta_0e^{-\alpha x_2}\hat{x}_2$, $\vec{B}=\alpha\beta_1e^{-\alpha x_2}\hat{x}_3$ (ii) $\vec{E}=\frac{\beta_0^{'}}{x_2^2}\hat{x}_2$, $\vec{B}=\frac{\beta_1^{'}}{x_2^2}\hat{x}_3$, and (iii) $\vec{E}=\frac{2\beta_0^{'}}{x_2^3}\hat{x}_2$, $\vec{B}=\frac{2\beta_1^{'}}{x_2^3}\hat{x}_3$, are studied. All these fields are generated from a systematic study of a particular type of differential equation whose coefficients are linear in independent variable. The Laplace transform approach is used to find the solutions and the corresponding eigenfunctions are expressed in terms of the hypergeometric functions $\,_{1}F_{1}(a', b'; x)$ for first two cases of the above configurations while the same are expressed in terms of the Bessel functions of first kind, $J_{n}(x)$, for the last case