Page 226 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 14: Classification of Partial Differential Equations

Notes
z z
2 = 3 ...(20)
x y 1
Now differentiating (20) by x

2 z 2 z
2 = 3 ( 2) 6 ...(21)
x 2 x y 1 1

And differentiating (20) by y

2 z 2 z
2 = 3 (1) ...(22)
x y y 2 1

Now eliminating 1 from (21) and (22) we have

2 z 2 z 2 z 2 z
2 2 4 2 2 = 0
x x y x y y

2 z 2 z 2 z
or 2 2 5 2 2 = 0 ...(23)
x x y y

Notes One can see that if there are two unknown functions in the relation between x, y and
z then we obtain second order partial differential equation.

Self Assessment

1. Set up the partial differential equation by treating z as dependent variable and x, y as
independent variables from the following relation
z f 1 (y x ) f 2 (y x )

2. Set up the partial differential equation from the following relation by treating z as
dependent variable and x, y as independent variable

e 5x 5z tan(y 3 ) ,(y 3 ) 0
x
x
14.3 Various Classes of Partial Differential Equations

In this section we shall discuss some partial differential equations that occur in problems or
propagation of waves in metals or strings, in electrostatics and gravitation, conduction of heat
and diffusion of things in certain media. The partial differential equations discussed in the last
two sections are generally partial differential equations. There are certain partial differential
equations which are of second order in nature or of higher order. Let us define the partial
derivatives of the dependent variable z of two independent variables x and y as

z z 2 z 2 z 2 z
, p , q , r s and 2 . t
x y x 2 x y y

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