The Laplace transform method for a scalar differential equation with symmetry/reflection

This work investigated a scalar differential equation involving proportional arguments of the form ${\varphi }^{\prime }(t)=\alpha \varphi (\gamma t)+\beta \varphi (-\gamma t)$. The terms $\varphi (\gamma t)$ and $\varphi (-\gamma t)$ represented proportional delay and reflection (time-reversal) effects, respectively, which arose in models exhibiting scaling and symmetry properties. A constructive analytical approach based on the Laplace transform was developed to derive a series representation of the solution. The method converted the original scalar equation into a recursive sequence of algebraic relations in the Laplace variable, allowing the systematic computation of successive terms. Explicit formulas for the series components were obtained, revealing a clear structure separating even and odd contributions. The convergence of the resulting series was rigorously established, showing that the solution existed globally and defined an entire function. Furthermore, the series representation was summed into a closed form expressed through hyperbolic functions, providing a compact analytical expression of the solution. Numerical illustrations confirmed the effectiveness of the derived formulation.