If a differential equation contains one or more partial derivatives, it is called partial differential equation Or PDE. Partial derivatives occur only when there are two or more than two independent variables. So in partial differential equations there are more than one independent variables. In what follows, x and y will, usually, be taken as the independent variables and z, the dependent variable so that z = f(x,y). Also the following notations would hold:
Now we shall study the ways in which PDE are formed. Then and extension of this method can be use to find solutions of special types of first and second order partial differential equation.
Formation of partial differential equations:
The PDE can be formed by any of the two methods, namely,
(i) by the elimination of the arbitrary constants
(ii) by the elimination of arbitrary functions
The complete solution of a first order PDE w ould contain one arbitrary function and that of the second order PDE would contain two arbitrary functions. Thus, in general, a complete solution of a PDE of nth order will contain n arbitrary functions.
Linear partial differential equation of the first order:
A linear PDE of the first order, commonly known as Lagrange’s linear equation is of the form
Pp + Qq = R
Where, P,Q and R are functions of x, y, z.
Such an equation is obtained by eliminating an arbitrary function f from f(u,v) = 0, where u and v are some function of x,y,z. We follow the following steps:
(a)Form the auxiliary equations : dx/P = dy/Q = dz/R
(b) Solve these simultaneous equations by method described earlier giving u = a and v = b as it solution.
(c) Write the solution as f(u,v) = 0 or u = f(v)
Illustrative example 1: Form a PDE by eliminating a the function from
