20.3 GENERAL AND PARTICULAR SOLUTIONS
If we take, for example,h(x, y)=expy, which clearly satisfies (20.15), then the general
solution is
u(x, y)=(expy)f(xexp(−^12 y)).Alternatively,h(x, y)=x^2 also satisfies (20.15) and so the general solution to the equation
can also be written
u(x, y)=x^2 g(xexp(−^12 y)),wheregis an arbitrary function ofp; clearlyg(p)=f(p)/p^2 .
20.3.2 Inhomogeneous equations and problemsLet us discuss in a more general form the particular solutions of (20.13) found
in the second example of the previous subsection. It is clear that, so far as this
equation is concerned, ifu(x, y) is a solution then so is any multiple ofu(x, y)or
any linear sum of separate solutionsu 1 (x, y)+u 2 (x, y). However, when it comes
to fitting the boundary conditions this is not so.
For example, althoughu(x, y) in (20.14) satisfies the PDE and the boundaryconditionu(1,y)=2y+ 1, the functionu 1 (x, y)=4u(x, y)=8xy+ 4, whilst
satisfying the PDE, takes the value 8y+4 on the linex= 1 and so does not satisfy
the required boundary condition. Likewise the functionu 2 (x, y)=u(x, y)+f 1 (x^2 y),
for arbitraryf 1 , satisfies (20.13) but takes the valueu 2 (1,y)=2y+1+f 1 (y)on
the linex= 1, and so is not of the required form unlessf 1 is identically zero.
Thus we see that when treating the superposition of solutions of PDEs twoconsiderations arise, one concerning the equation itself and the other connected
to the boundary conditions. Theequationis said to be homogeneous if the fact
thatu(x, y) is a solution implies thatλu(x, y), for any constantλ, is also a solution.
However, theproblemis said to be homogeneous if, in addition, the boundary
conditions are such that if they are satisfied byu(x, y) then they are also satisfied
byλu(x, y). The last requirement itself is referred to as that ofhomogeneous
boundary conditions.
For example, the PDE (20.13) is homogeneous but the general first-orderequation (20.9) would not be homogeneous unlessR(x, y) = 0. Furthermore,
the boundary condition (i) imposed on the solution of (20.13) in the previous
subsection is not homogeneous though, in this case, the boundary condition
u(x, y) = 0 on the liney=4x−^2would be, sinceu(x, y)=λ(x^2 y−4) satisfies this condition for anyλand, being a
function ofx^2 y, satisfies (20.13).
The reason for discussing the homogeneity of PDEs and their boundary condi-tions is that in linear PDEs there is a close parallel to the complementary-function
and particular-integral property of ODEs. The general solution of an inhomo-
geneous problem can be written as the sum ofanyparticular solution of the
problem and the general solution of the corresponding homogeneous problem (as