A freeform reflector design method, which is mainly based on a first-order linear partial differential equation, is proposed for uniform rectangular illuminance distribution in the field of LED illumination. The interaction between the freeform surface and the light beam is depicted based on theory of the differential geometry and Snell′s law. The energy topological relation between the Lambertian luminaire and the illuminated rectangular surface is established according to the LED luminous intensity distribution. The method deducts a first-order linear partial differential equation with some boundary conditions to represent the freeform reflector. The boundary conditions and the partial differential equation are solved by the Runge-Kutta method and finite difference method, respectively. The numerical results are validated in the form of raytracing, which reveal that the luminous flux efficiency is about 94%, the transverse uniformity of illuminance on the target surface is 0.9 and the longitudinal uniformity of illuminance on the target surface is 0.8. The numerical computation time is less than 1 s.