An Elementary Course on Partial Differential Equations

In Chapters 4 and 5, we have discussed various methodsfor finding the solutions of first- and second-orderpartial differential equations (PDEs). Now, weundertake the solution of those linear PDEs, inwhich higher order derivatives occur. Thetheoretical framework and methods of solutiondeveloped for first and second-order linearequations are extendable to a very limited class oflinear equations of higher (arbitrary) order.Generally, it is very difficult to solve any linearPDE, wherein the coefficients are functions ofx and y. Henceforth, we start ourjourney with the description of certain kinds oflinear PDEs, in which the various terms aremultiplied by constants only. We devote Sections 6.1and 6.2 to such linear PDEs. In Sections 6.3–6.5, wediscuss some special types of linear equations withvariable coefficients, which are reducible easily toequations with constant coefficients and hencebecome solvable.

6.1 Homogeneous Linear PDEs with ConstantCoefficients

The most general linear PDE with constant coefficientsof order n is of theform

where the coefficients A0,A1,…,An, B0,…, P, Q, Z all are constantsand A0,A1,…,An do not vanishsimultaneously. Recall that an equation of the form(6.1) is called homogeneous whenever f(x,y) = 0. Otherwise, it is callednon-homogeneous.

Conveniently, Eq. (6.1) can be symbolically writtenas

where the polynomial

remains a differential operator with constantcoefficients.

As mentioned in Chapter 2, the mixed partialderivatives of zinvolved in a PDE are assumed to be independent ofthe order of differentiation. Subsequently, for anequation F(D, D')z = f(x, y)with constant coefficients such that F(D,D') = g(D, D') ⋅h(D, D'), we have h(D,D') ⋅ g(D, D') =F(D, D'). This observation canbe described in the form of the following resultthat has an important bearing in the context oflinear PDEs.

Proposition 6.1.Let F(D, D')z = f(x, y)be a linear PDE with constantcoefficients. If the polynomial F(D, D') can be decomposed into some factors, then theorder in which these factors occur isunimportant.

In this section, we discuss the techniques for solvingthe homogeneous linear PDEs with constantcoefficients of arbitrary order. For the sake ofbrevity, we divide this section into foursubsections.