Monday Math 17

Let us consider the partial differential equation  for unknown function of two variables .
First, we see this is simple to find a general solution for this equation. We have


and thus integrating with respect to x, we find:
, for some function h(y) of y only.
Now, we integrate that with respect to y to get:
, with f(x) some function of x only, and . This is the general solution to the equation.

Now, however, let us consider the change of variables  given by
, ,
where c>0 is some constant. From these we find:
, 
, 

Then we see by the chain rule for partial derivatives that

and thus that


And thus our differential equation  in terms of the new coordinates is thus:




This is the one-dimensional wave equation with speed c (see also this ‘Physics Friday’ post). Now recall that we found the general solution to our original equation  to be . Now, inverting our coordinate transformation , , we see , , and so the general solution to

is , for functions of a single variable f and g.
[Note that the physical interpretation of this is that  is a wave moving rightward (+x direction), and  is a wave moving leftward (-x direction).]