20.3 GENERAL AND PARTICULAR SOLUTIONS
equations by cross-multiplication, obtaining
∂p
∂y∂ui
∂x=∂p
∂x∂ui
∂y,or, for our specific form,p=x^2 +2y,
∂ui
∂x=x∂ui
∂y. (20.8)
It is thus apparent that not only are the three functionsu 1 ,u 2 u 3 solutions of the
PDE (20.8) but so also isany arbitrary functionf(p) of which the argumentphas
the formx^2 +2y.
20.3 General and particular solutionsIn the last section we found that the first-order PDE (20.8) has as a solutionany
function of the variablex^2 +2y. This points the way for the solution of PDEs
of other orders, as follows. It isnotgenerally true that annth-order PDE can
always be considered as resulting from the elimination ofnarbitraryfunctions
from its solution (as opposed to the elimination ofnarbitraryconstantsfor an
nth-order ODE, see section 14.1). However, given specific PDEs we can try to
solve them by seeking combinations of variables in terms of which the solutions
may be expressed as arbitrary functions. Where this is possible we may expectn
combinations to be involved in the solution.
Naturally, the exact functional form of the solution for any particular situationmust be determined by some set of boundary conditions. For instance, if the PDE
contains two independent variablesxandythen for complete determination of
its solution the boundary conditions will take a form equivalent to specifying
u(x, y) along a suitable continuum of points in thexy-plane (usually along a line).
We now discuss the general and particular solutions of first- and second-order PDEs. In order to simplify the algebra, we will restrict our discussion
to equations containing just two independent variablesxandy. Nevertheless,
the method presented below may be extended to equations containing several
independent variables.
20.3.1 First-order equationsAlthough most of the PDEs encountered in physical contexts are second order
(i.e. they contain∂^2 u/∂x^2 or∂^2 u/∂x∂y, etc.), we now discuss first-order equations
to illustrate the general considerations involved in the form of the solution and
in satisfying any boundary conditions on the solution.
The most general first-order linear PDE (containing two independent variables)