Diffusion probabilistic models (DPMs) are a key component in modern generative models. DPM-solvers have achieved reduced latency and enhanced quality significantly, but have posed challenges to find the exact inverse (i.e., finding the initial noise from the given image). Here we investigate the exact inversions for DPM-solvers and propose algorithms to perform them when samples are generated by the first-order as well as higher-order DPM-solvers. For each explicit denoising step in DPM-solvers, we formulated the inversions using implicit methods such as gradient descent or forward step method to ensure the robustness to large classifier-free guidance unlike the prior approach using fixed-point iteration. Experimental results demonstrated that our proposed exact inversion methods significantly reduced the error of both image and noise reconstructions, greatly enhanced the ability to distinguish invisible watermarks and well prevented unintended background changes consistently during image editing.