Origins of the [partial differential] equations
The diffusion equation arises in physics from the evolution of temperature or of gas density. The wave equation represents the propagation of linear waves, for example along a compressible spring. The sine-Gordon equation represents nonlinear waves obtained for example as the limit of a very large number of pendulums all connected to a spring. The traditional name of the equation is a pun on the Klein–Gordon equation that appears in relativistic quantum mechanics and in describing strings in elastic media. It is notable that unlike with ODEs, essentially all PDEs that have been widely studied come quite directly from physics. My PDE on page 165 is however an exception.