Page 224 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 14: Classification of Partial Differential Equations

Notes
u u u
x y = 0
x y z
is a first order partial differential equation involving three variables. So in these units involving
partial differential equations we may have to deal with first order, second order or higher order
partial differential equations.
14.2 Derivation of Partial Differential Equations

Example 1: Let us form the differential equation from the relation
2
2
2
lx + my + nz = (x + y +z ) ...(5)
Differentiating equation partially with respect to x and y
z dz
l n = (x 2 y 2 z 2 ) 2x 2z ...(6)
x dx

z 2 2 2 z
and m n = (x y z ) 2y 2z ...(7)
y y
Eliminating
z z
l n x z
x x
=
z z
m n y z
y y

z z z z z
or (l np )y m n x z l n z m n = 0
y y x x y

or (l np )y (m nq )x z (lq mp ) = 0 ...(8)

Notes When the relation like (6) contains more than one function partial differential
equations of the higher order will be obtained.

Example 2: Find the partial differential equation from the relation
x y
= ...(9)
z z
by treating z as dependent variable and x, y as independent variables.

Solution: Differentiating (9) with respect to x, we have

1 x y
p = 2 p ...(10)
z z 2 z

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