The Need For the Method of Characteristics
Consider the functionwhereis an arbitrary function. It is easier to understand what is happening if we define
The partial derivatives, on using the chain rule, are
Thereforefor any functionThis illustrates an important point. Ordinary differential equations have arbitrary constants but partial differential equations have arbitrary
functions. The functionis determined by an initial condition.
Assume that the initial condition, atiswitha given function. If
Hence, the unknown function,is the given functionbut the argument is replaced by hence, the solution is
Ifthen satisfies the equation and the initial condition.
Example: Consider the linear equation
The solution iswhereis an arbitrary function. The partial derivatives are
Henceand the equation is satisfied.
Example: Consider the linear equation with non-constant coefficients,
The solution is given bywhereis again an arbitrary function. Denoting
andand so the equation is clearly satisfied.
The argument of the function f is determined using the method of characteristics.