The Cahn-Hilliard equation is a fundamental model in the study of phase separation and coarsening phenomena in binary mixtures. The paper investigates the perturbative solutions of the one-dimensional Cahn-Hilliard equation for small spatial and temporal variables. Starting with a uniform state, a small perturbation was introduced and first-order perturbation expansion was derived. Utilizing Fourier transforms, the linearized form of Cahn-Hilliard equation was solved to obtain the general solution. The dispersion relation revealed the growth rates of perturbation modes, providing insight into the early-time dynamics of phase separation. The analytical approach lays the groundwork for understanding the evolution of small perturbations and their impact on the phase separation process in binary systems. This work has potential applications in materials science, particularly in understanding the microstructural development of alloys and polymer blends.