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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