In this article, we construct the general solution to a nonlocal linear differential equation of second-order: \(\begin{aligned} y^{\prime \prime }(t)=\gamma y^{\prime }(t)+\delta y^{\prime }(-t)+\alpha y(t)+\beta y(-t)+f(t),\, t\in {\mathbb {R}}\, , \end{aligned}\) where \(\alpha , \beta , \gamma , \delta \) are arbitrary real constants and f is continuous. To address the nonlocal model, we adopt an innovative strategy by representing the solution as a combination of even and odd functions, converting the nonlocal model into a system of local models. Using standard techniques for ordinary differential equations, we derive the general solution for the original nonlocal model, characterized by two arbitrary constants. We explore five main cases and their subcases based on the coefficients. Our investigation reveals that the Cauchy problem for nonlocal differential equations exhibits complex solution behavior, where the traditional existence and uniqueness theorem doesn’t always apply, especially due to the coefficients’ characteristics. Additionally, we present three examples with their corresponding general solutions.
