Notice that where the solution of an ODE contains arbitrary constants, the solution to a PDE contains arbitrary functions. In the same spirit, while an ODE of order m has m linearly independent solutions, a PDE has infinitely many (there are arbitrary functions in the solution). These are consequences of the fact that a function of two variables contains immensely more (a whole dimension worth) of information than a function of only one variable.
The method of characteristics is a powerful method that allows one to reduce any first-order linear PDE to an ODE, which can be subsequently solved using ODE techniques. We will see in later lectures that a subclass of second order PDEs second order hyperbolic equations can be also treated with a similar characteristic method.
We derived the wave and heat equations from physical principles, identifying the unknown function with the amplitude of a vibrating string in the first case and the temperature in a rod in the second case. Understanding the physical significance of these PDEs will help us better grasp the qualitative behaviour of their solutions, which will be derived by purely mathematical techniques in the subsequent lectures. The physicality of the initial and boundary conditions will also help us immediately rule out solutions that do not conform to the physical laws behind the appropriate problems.
The second order linear PDEs can be classified into three types, which are invariant under changes of variables. The types are determined by the sign of the discriminate. This exactly corresponds to the different cases for the quadratic equation satisfied by the slope of the characteristic curves. We saw that hyperbolic equations have two distinct families of (real) characteristic curves, parabolic equations have a single family of characteristic curves, and the elliptic equations have none. All the three types of equations can be reduced to canonical forms. Hyperbolic equations reduce to a...