An Elementary Course on Partial Differential Equations

There are several ways of solving partial differentialequations (PDEs), which are utilised in specificsituations depending on the order of the underlyingequation. In succeeding chapters, we discuss severalPDEs of order one (Chapter 4), two (Chapter 5), andhigher (Chapter 6) in a systematic manner.Nevertheless, we begin our journey with some basicmethods to solve special types of PDEs, which aresolvable easily. These basic methods often do notdepend on the order of PDEs. In this chapter, first,we undertake certain types of PDEs, which can besolved merely by integration. Thereafter, weconsider the special classes of linear PDEs, whichcan be solved on the lines of linear ordinarydifferential equations (ODEs). Finally, we describethe method of separation of variables.

3.1 Direct Integration Method

Roughly speaking, the operations of ‘differentiation’and ‘integration’ are the inverses of each other. Toassociate an inverse of partial differentiation,consider a function u(x, y),whose partial derivative w.r.t. x is f(x, y),i:e:,

If the above relation is satisfied, then we say thatu(x, y) is a partial integralof f(x, y) w.r.t. x and symbolically, wewrite

In order to have a substitute of constant ofintegration, we replace an arbitrary function𝜙 of y owing to the fact that anarbitrary constant naturally is absorbed with𝜙(y) so that we have

which amounts to saying that if ux(x,y) = f(x, y),then

This motivates us to define the concept of partialintegration.

Partial Integration: Let f(x1, x2,…, xn) be a function of n independent variables,then the partial integral of f w.r.t. an independent variable xi (1 ≤ i ≤ n), often denoted by ∫ f(x1, x2,…, xn)𝜕xi, is defined as theintegral of f w.r.t.xitreating the rest n -1 independent variables as constants and taking anarbitrary function 𝜙(x1, x2,…, xi-1, xi+1,…, xn) of these n - 1 variables instead ofarbitrary constant of integration.

Generally, a PDE containing single partial derivativecan be easily solved by the technique of directintegration. Such PDEs are illustrated by Examples3.1–3.3. On the other hand, a PDE involving twoderivatives of consecutive orders in its distinctterms and not containing dependent variableexplicitly, sometimes can also be solved by thismethod (see Examples 3.4–3.6).