ODENet is a deep neural network architecture in which a stacking structure of ResNet is implemented with an ordinary differential equation (ODE) solver. It can reduce the number of parameters and strike a balance between accuracy and performance by selecting a proper solver. It is also possible to improve the accuracy while keeping the same number of parameters on resource-limited edge devices. In this paper, using Euler method as an ODE solver, a part of ODENet is implemented as a dedicated logic on a low-cost FPGA (Field-Programmable Gate Array) board, such as PYNQ-Z2 board. Two variants, one for high accuracy and the other for performance, are proposed and implemented on the FPGA board as well. They are evaluated in terms of parameter size, accuracy, execution time, and resource utilization on the FPGA. The results show that an overall execution time of ODENet and its variants is improved by up to 2.50 times compared to a pure software execution when a part of convolution layers is executed by the programmable logic.