Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.3 GENERAL AND PARTICULAR SOLUTIONS

homogeneous equation isu(x, y)=f(x^2 +y^2 ) for arbitrary functionf. Now by inspection
a particular integral of (20.18) isu(x, y)=− 3 y, and so the general solution to (20.18) is

u(x, y)=f(x^2 +y^2 )− 3 y.
Boundary condition (i) requiresu(x,0) =f(x^2 )=x^2 ,i.e.f(z)=z, and so the particular
solution in this case is

u(x, y)=x^2 +y^2 − 3 y.

Similarly, boundary condition (ii) requiresu(1,0) =f(1) = 2. One possibility isf(z)=2z,
and if we make this choice, then one way of writing the most general particular solution
is

u(x, y)=2x^2 +2y^2 − 3 y+g(x^2 +y^2 ),

wheregis any arbitrary function for whichg(1) = 0. Alternatively, a simpler choice would
bef(z) = 2, leading to

u(x, y)=2− 3 y+g(x^2 +y^2 ).

Although we have discussed the solution of inhomogeneous problems only

for first-order equations, the general considerations hold true for linear PDEs of

higher order.

20.3.3 Second-order equations

As noted in section 20.1, second-order linear PDEs are of great importance in

describing the behaviour of many physical systems. As in our discussion of first-

order equations, for the moment we shall restrict our discussion to equations with

just two independent variables; extensions to a greater number of independent

variables are straightforward.

The most general second-order linear PDE (containing two independent vari-

ables) has the form

A

∂^2 u

∂x^2

+B

∂^2 u

∂x∂y

+C

∂^2 u

∂y^2

+D

∂u

∂x

+E

∂u

∂y

+Fu=R(x, y), (20.19)

whereA,B,...,FandR(x, y) are given functions ofxandy. Because of the nature

of the solutions to such equations, they are usually divided into three classes, a

division of which we will make further use in subsection 20.6.2. The equation

(20.19) is calledhyperbolicifB^2 > 4 AC,parabolicifB^2 =4ACandellipticif

B^2 < 4 AC. Clearly, ifA,BandCare functions ofxandy(rather than just

constants) then the equation might be of different types in different parts of the

xy-plane.

Equation (20.19) obviously represents a very large class of PDEs, and it is

usually impossible to find closed-form solutions to most of these equations.

Therefore, for the moment we shall consider only homogeneous equations, with

R(x, y) = 0, and make the further (greatly simplifying) restriction that, throughout

the remainder of this section,A,B,...,Fare not functions ofxandybut merely

constants.