Solving a non-local linear differential equation model of the Newtonian-type

Motivated by recent studies on non-local integrable models, we consider a non-local inhomogeneous linear differential equation model of Newtonian type:

$$\begin{aligned} \hspace{42pt}x''(t)=\lambda x(t)+\mu x(-t) +f(t),\ t\in {\mathbb {R}}, \end{aligned}$$

where \(\lambda \) and \(\mu \) are real constants and f is continuous. Through decomposing functions into their even and odd parts, we transform the non-local model into a local model, and then with the classical ODE technique, solve the resulting local model under the even and odd constraints. The general solution involving two arbitrary constants is presented in nine cases of the coefficients.