Introduction 19
1.6 Separation of Variables for Linear, Homogeneous PDEs
In the mid-1700s, Daniel Bernoulli and, later, Jean le Rond d’Alembert exper-
imented with a new technique for producing solutions of linear, homogeneous
PDEs. This method, calledseparation of variables,§entails the reduction
of a PDE to an ODE (or, more commonly, to a number of ODEs, each corre-
sponding to a different independent variable), a recurrent theme in the study
of PDEs.
Definition 1.4Given a PDE inu=u(x, y), we say thatuis aproduct
solutionif
u(x, y)=f(x)g(y)
for functionsf andg. More generally, u=u(x 1 ,x 2 ,...,xn)is a product
solution of a PDE in thenvariablesx 1 ,x 2 ,...,xnif
u(x 1 ,x 2 ,...,xn)=f 1 (x 1 )f 2 (x 2 )...fn(xn)for functionsf 1 ,f 2 ,...,fn.(SeeExercise23.)
In practice, it is more common to writeu(x, y)=X(x)Y(y),u(x, y, z)=
X(x)Y(y)Z(z), etc.
How does the method work? Let’s look at some examples.
Example 1Find all product solutions of the first-order, linear, homogeneous
PDEux+uy=0.
We search for all solutions of the formu(x, y)=X(x)Y(y). Using the facts
that
ux=X′Y and uy=XY′,
we substitute into the PDE and get
X′Y+XY′=0. (1.13)How does this help us? Well, a little algebra (specifically, dividing both sides
byXY¶)givesus
X′
X
=−
Y′
Y
, (1.14)
§When studying a linear, homogeneous PDE, the first question that a mathematician usu-
ally asks is, “Is the equation separable?”
¶Of course, if eitherXorYis the zero-function, then we may not divide byXY. However,
in this case,uis the zero-function, which is already known to be a solution toanylinear,
homogeneous PDE. “Officially,” we may use this method only on two-dimensional regions
whereX(x)Y(y)= 0 although, in practice, this turns out not to be an issue.