General form of solution

PDES: GENERAL AND PARTICULAR SOLUTIONS
describes the quantum mechanical wavefunction u(r, t) of a non-relativistic particle
of mass m; is Planck’s constant divided by 2π. Like the diffusion equation it is
second order in the three spatial variables and first order in time.
20.2 General form of solution
Before turning to the methods by which we may hope to solve PDEs such as
those listed in the previous section, it is instructive, as for ODEs in chapter 14, to
study how PDEs may be formed from a set of possible solutions. Such a study
can provide an indication of how equations obtained not from possible solutions
but from physical arguments might be solved.
For definiteness let us suppose we have a set of functions involving two
independent variables x and y. Without further specification this is of course a
very wide set of functions, and we could not expect to find a useful equation that
they all satisfy. However, let us consider a type of function ui (x, y) in which x and
y appear in a particular way, such that ui can be written as a function (however
complicated) of a single variable p, itself a simple function of x and y.
Let us illustrate this by considering the three functions
u1 (x, y) = x4 + 4(x2 y + y 2 + 1),
u2 (x, y) = sin x2 cos 2y + cos x2 sin 2y,
x2 + 2y + 2
.
u3 (x, y) = 2
3x + 6y + 5
These are all fairly complicated functions of x and y and a single differential
equation of which each one is a solution is not obvious. However, if we observe
that in fact each can be expressed as a function of the variable p = x2 + 2y alone
(with no other x or y involved) then a great simplification takes place. Written
in terms of p the above equations become
u1 (x, y) = (x2 + 2y)2 + 4 = p2 + 4 = f1 (p),
u2 (x, y) = sin(x2 + 2y) = sin p = f2 (p),
p+2
(x2 + 2y) + 2
=
= f3 (p).
u3 (x, y) =
3(x2 + 2y) + 5
3p + 5
Let us now form, for each ui , the partial derivatives ∂ui /∂x and ∂ui /∂y. In each
case these are (writing both the form for general p and the one appropriate to
our particular case, p = x2 + 2y)
dfi (p) ∂p
∂ui
=
= 2xfi ,
∂x
dp ∂x
dfi (p) ∂p
∂ui
=
= 2fi ,
∂y
dp ∂y
for i = 1, 2, 3. All reference to the form of fi can be eliminated from these
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