Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

for ODEs, we require that the particular solution is not already contained in the

general solution of the homogeneous problem). Thus, for example, the general

solution of

∂u

∂x

−x

∂u

∂y

+au=f(x, y), (20.16)

subject to, say, the boundary conditionu(0,y)=g(y), is given by

u(x, y)=v(x, y)+w(x, y),

wherev(x, y) is any solution (however simple) of (20.16) such thatv(0,y)=g(y)

andw(x, y) is the general solution of

∂w

∂x

−x

∂w

∂y

+aw=0, (20.17)

withw(0,y) = 0. If the boundary conditions are sufficiently specified then the only

possible solution of (20.17) will bew(x, y)≡0andv(x, y) will be the complete

solution by itself.

Alternatively, we may begin by finding the general solution of the inhomoge-

neous equation (20.16)withoutregard for any boundary conditions; it is just the

sum of the general solution to the homogeneous equation and a particular inte-

gral of (20.16), both without reference to the boundary conditions. The boundary

conditions can then be used to find the appropriate particular solution from the

general solution.

We will not discuss at length general methods of obtaining particular integrals

of PDEs but merely note that some of those methods available for ordinary

differential equations can be suitably extended.§

Find the general solution of

y

∂u

∂x

−x

∂u

∂y

=3x. (20.18)

Hence find the most general particular solution(i)which satisfiesu(x,0) =x^2 and(ii)which

has the valueu(x, y)=2at the point(1,0).

This equation is inhomogeneous, and so let us first find the general solution of (20.18)
without regard for any boundary conditions. We begin by looking for the solution of the
corresponding homogeneous equation ((20.18) but with the RHS equal to zero) of the
formu(x, y)=f(p). Following the same procedure as that used in the solution of (20.13)
we find thatu(x, y) will be constant along lines of (x, y)thatsatisfy

dx

y

=

dy

−x

⇒

x^2

2

+

y^2

2

=c.

Identifying the constant of integrationcwithp/2, we find that the general solution of the

§See for example H. T. H. Piaggio,An Elementary Treatise on Differential Equations and their

Applications(London: G. Bell and Sons, Ltd, 1954), pp. 175 ff.