## The Need For the Method of Characteristics

Consider the function where is an arbitrary function. It is easier to understand what is happening if we define The partial derivatives, on using the chain rule, are and Therefore for any function This illustrates an important point. Ordinary differential equations have arbitrary constants but partial differential equations have arbitrary

functions. The function is determined by an initial condition.

Assume that the initial condition, at is with a given function. If then Hence, the unknown function, is the given function but the argument is replaced by hence, the solution is If then satisfies the equation and the initial condition.

Example: Consider the linear equation The solution is where is an arbitrary function. The partial derivatives are and Hence and the equation is satisfied.

Example: Consider the linear equation with non-constant coefficients, The solution is given by where is again an arbitrary function. Denoting  and and so the equation is clearly satisfied.

The argument of the function f is determined using the method of characteristics. 