Introduction 7
Since
u=f(y) (1.3)
represents all possible solutions of (1.1), we call (1.2) thegeneral solutionof
(1.1). So, where the general solution of an ODE involves arbitrary constants,
the general solution of a PDE involvesarbitrary functions.
With these ideas in mind, there already areplentyof PDEs we can solve.
Example 1Find all solutionsu=(x, y)ofux=x^2 +y^2 .Wehave
u=∫
(x^2 +y^2 )dx=x^3
3+xy^2 +f(y),wheref(y) is an arbitrary function ofy.
Example 2Find the general solution ofuy=xz+yz.Wehave
u=∫
(xz+yz)dy=xyz+y^2 z
2+f(x, z),wherefis an arbitrary functionofxandz.
We need not restrict ourselves to equations of the first order.Example 3Find the general solution ofuxx=12xy. We integrate twice, of
course:
ux=∫
12 xy dx=6x^2 y+f(y),wherefis an arbitrary function ofy,then
u=∫
(6x^2 y+f(y))dx=2x^3 y+xf(y)+g(y),wheregis an arbitrary function ofy.
Example 4Do the same foruxy=cosx. First, we have
ux=∫
cosxdy=ycosx+f(x).Then,
u=∫
(ycosx+f(x))dx.Now, what is
∫
f(x)dx? If we antidifferentiatef(x) with respect tox,wejust
get another function ofx. However, we also get an “arbitrary constant,” that
is, in this case, an arbitrary function ofy.So
∫
f(x)dx=f 1 (x)+g(y)