Popularity prediction for information cascades is a fundamental and challenging task. While most existing methods consider this as a discrete problem, we argue that popularity trajectory prediction is more practical, as it aims to forecast the entire continuous trajectory of how popularity unfolds over arbitrary future time. However, traditional methods for popularity trajectory prediction primarily rely on specific diffusion mechanism assumptions, which may not align well with real-world dynamics, limiting their performance. To address this limitation, we propose NODEPT, a novel approach based on neural ordinary differential equations (ODEs) for popularity trajectory prediction. We first employ an encoder to initialize the latent state representations that capture the co-evolution structural characteristics and temporal patterns of cascades. 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.