Popularity prediction for information cascades has significant applications across various domains, including opinion monitoring and advertising recommendations. While most existing methods consider this as a discrete problem, popularity actually evolves continuously, exhibiting rich dynamic properties such as change rates and growth patterns. In this paper, we argue that popularity trajectory prediction is more practical, as it aims to forecast the entire trajectory of how popularity unfolds over arbitrary future time. This approach offers insights into both instantaneous popularity and the underlying dynamic properties. However, traditional methods for popularity trajectory prediction primarily rely on specific diffusion mechanism assumptions, which may not align well with real-world dynamics and compromise their performance. To address these limitations, we propose NODEPT, a novel approach based on neural ordinary differential equations (ODEs) for popularity trajectory prediction. NODEPT models the continuous dynamics of the underlying diffusion system using neural ODEs. We first employ an encoder to initialize the latent state representations of information cascades, consisting of two representation learning modules that capture the co-evolution structural characteristics and temporal patterns of cascades from different perspectives. More importantly, we then introduce an ODE-based generative module that learns the dynamics of the diffusion system in the latent space. Finally, a decoder transforms the latent state into the prediction of the future popularity trajectory. Our experimental results on three real-world datasets demonstrate the superiority and rationality of the proposed NODEPT method.