Based on the conditions of aplanatism, a differential equation design method has been applied to the optimization of finite-conjugate, multimirror systems for soft x-ray applications. The shapes of two surfaces in a multimirror system are determined by numerically solving differential equations for the mirror surfaces. The shapes of other mirrors in the system are used to control the remaining aberrations of those systems, such as astigmatism, field curvature, and distortion. Applications include a three-minor projection-lithography system, which has been optimized by this method and compared with results obtained by conventional optical design, and a two-mirror soft x-ray imaging microscopy system, which is predicted to yield a diffraction-limited performance for a large N.A. © 1993 Optical Society of America.