PARTIAL DIFFERENTIAL EQUATIONS µx µy and

PARTIAL DIFFE RENTIAL E QUAT IONS Partial differential equations are those equations which

contain partial derivatives,independent Variables and dependent variables.

The independent variables will be denoted by `x' & `y' and dependent variable by `z'. The Partial differential coefficients are denoted as follows

=r,

O RDER : Order of partial differential equations is the same as that

of the highest differential coefficients in it.

ME TH ODS O F FO R MING PART IAL DIFFE R ENTIAL E QUAT IO N

The PDE'S formedby 2 methods

1) By eliminating arbitrary constants

2) By eliminating

arbitrary functions

Methods of eliminating arbitrary constants

Ex;-form of PDE from

x2 +y2+ (z-c) 2=a2

X2+y2+ (z-c) 2=a2

................> (1)

The

above equation contains 2 arbitrary constants a & c

Differentiate equation (1) partial w.r.t, x we get

2x+2(z-c)

X+ (z-c) p=0 .......> (2)

Differentiate (1)

partially w.r.t, ywe get

2y+2(z-c)

Y+ (z-c) q=0 .........> (3)

Eliminating c from (2) and (3)

(2).....>

z-c=-x/p

Substituting above value in (3) we get

Y-x/p*q=0

Y*p-x*q=0

Methods of eliminating of arbitrary functions

Ex;-form of PDE fromz=f(x2-y2)

z=f(x2-y2)

..........> (1)

Differentiate (1)

partially

w.r.t, x and y we get

p=

'(x2-y2) 2x .....> (2)

q=

'(x2-y2) (-2y) ......>(3)

Dividing (2) and (3) we get P/q=-x/y (or) p*y=-q*x (Y*p)+(x*q)=0

L AG RANG E'S L INEAR E QUAT IO N IS AN E QUATIO N O F TH E T YPE P*p +Q *q= R . wher e P, Q , R a r e th e function s of x, y, z a nd

p=

, q=

Proof; - P* p + Q* q= R .........> (1) The form of the equation is obtained by eliminating an arbitrary

function f from F (u, v) = 0........> (2) Where u, v, w, are the functions of x, y, z.

Differentiating (2) partially w .r. t, x and y

.......> (3)

.......> (4)

Let us eliminate From (3), (

from (3) and (4) .......> (5)

From (4),

Dividing (5) and (6) we get (

(7)

If (1) and (7) are the same then coefficients p, q is equal

P=uy

?

v z

u z

?

v z

Q=uz

?

v x

R=ux

?

v y

u x

?

v z

u y

?

v x

........>

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