Page 228 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 14: Classification of Partial Differential Equations

where c , c , ... c are constant of x and y. Notes
1 2 6
By means of transformations we can reduce the homogeneous partial differential equations into
those with constant coefficients.

Self Assessment

3. Classify the equation

2 2 2
z 3 z z 0
x 2 x y y 2

into one of the categories i.e. elliptical, hyperbolic or parabolic type.
4. Reduce the equation

2 z 2 z 2 z
x 2 2 2xy y 2 2 0
x x y y
to equation with constant coefficients.

14.4 Summary

 Like ordinary differential equations partial differential equations play an important part
in understanding certain processes.
 There are various types of partial equations like partial differential equations of first
order. It involves only first partial derivatives of the dependent variable.
 Then there are partial differential equations of second or higher order and involve higher
order than the first one, derivatives of the dependent variables.
 The most important second order partial differential equations can be either elliptic or
parabolic or hyperbolic and play important role in most physical problems.

 In the subsequent units various methods will be given to tackle these types of equations.
14.5 Keyword

The classification of partial differential equations help us to choose appropriate method for
solving these partial differential equations.

14.6 Review Questions

1. Set up partial differential equations by eliminating the constants a and b:
y 2 (x a ) 2 y 2 2z b

2. Set up partial differential equation by eliminating b and a from the following equation

2
z ax 3a y b
3. Reduce the following equation to an equation having constant coefficients of its derivatives
2 z 2 z 2 z z
3 4
x 2 4xy 4y 2 6y x y
x 2 x y y 2 y

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