1 First-Order Equations and Characteristics

A partial differential equation relates an unknown function of several variables to its partial derivatives. The great equations of mathematical physics are of this kind: the heat equation for diffusion, the wave equation for vibration and sound, Laplace’s equation for steady potentials, the Navier–Stokes equations for fluids, Maxwell’s equations for electromagnetism, Schrödinger’s equation for quantum mechanics. So are the Black–Scholes equation of finance and the equations of population dynamics in which age or space is a variable. This course develops the theory of the classical linear equations and the main methods for solving them, beginning with first-order equations, where a single idea, the method of characteristics, reduces the partial differential equation to ordinary differential equations along special curves.

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