Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSfor ODEs, we require that the particular solution is not already contained in thegeneral solution of the homogeneous problem). Thus, for example, the generalsolution of∂u∂x − x ∂u + au = f(x, y), (20.16)∂ysubject to, say, the boundary condition u(0,y)=g(y), is given byu(x, y) =v(x, y)+w(x, y),where v(x, y) is any solution (however simple) of (20.16) such that v(0,y)=g(y)and w(x, y) is the general solution of∂w∂x − x∂w + aw =0, (20.17)∂ywith w(0,y) = 0. If the boundary conditions are sufficiently specified then the onlypossible solution of (20.17) will be w(x, y) ≡ 0andv(x, y) will be the completesolution by itself.Alternatively, we may begin by finding the general solution of the inhomogeneousequation (20.16) without regard for any boundary conditions; it is just thesum of the general solution to the homogeneous equation and a particular integralof (20.16), both without reference to the boundary conditions. The boundaryconditions can then be used to find the appropriate particular solution from thegeneral solution.We will not discuss at length general methods of obtaining particular integralsof PDEs but merely note that some of those methods available for ordinarydifferential equations can be suitably extended. §◮Find the general solution ofy ∂u∂x − x ∂u =3x. (20.18)∂yHence find the most general particular solution (i) which satisfies u(x, 0) = x 2 and (ii) whichhas the value u(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 thecorresponding homogeneous equation ((20.18) but with the RHS equal to zero) of theform u(x, y) =f(p). Following the same procedure as that used in the solution of (20.13)we find that u(x, y) will be constant along lines of (x, y) thatsatisfydxy = dy−x⇒x22 + y22 = c.Identifying the constant of integration c with p/2, we find that the general solution of the§ See for example H. T. H. Piaggio, An Elementary Treatise on Differential Equations and theirApplications (London: G. Bell and Sons, Ltd, 1954), pp. 175 ff.686