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This Ph.D.-level course introduces the ordinary and partial differential equations theories from an applied mathematician’s viewpoint. The course starts with concepts widely used in applied mathematics and functional analysis, including Banach spaces, Hilbert spaces, distributions, Fourier transform, and Sobolev spaces. Then we discuss existence and uniqueness theory for weak solutions and viscosity solutions to ordinary and partial differential equations, with an emphasis on those that arise in mathematical physics, calculus of variations, or deterministic and stochastic control theory.