Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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 solutions

In 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 situation

must 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 equations

Although 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)