CHAPTER 1
PARTIAL DIFFERENTIAL EQUATIONS
A partial differential equation is an equation involving a function of two or more variables and some of its partial derivatives. Therefore a partial differential equation contains one dependent variable and one independent variable. Here z will be taken as the dependent variable and x and y the independent variable so that z f x, y . e will use the following standard notations to denote the partial derivatives. z z ! z ! z ! z p, q, r , s , ! t x y x y x ! y
The order of partial differential equation is that of the highest order derivative occurring in it.
Formation of partial differential eqation ! There are two methods to form a partial differential equation. "i# $y elimination of arbitrary constants. "ii# $y elimination of arbitrary functions.
Pro"lem#
Formation of partial differential eqation "$ elimination of ar"itrar$ %on#tant#%
&1'Form t(e partial differential eqation "$ eliminatin) t(e ar"itrar$ %on#tant# ! ! from z ax by a b *
Soltion!
&
! ! 'iven z ax by a b
(((((... "&#
Here we have two arbitrary constants a ) b. *ifferentiating equation "&# partially with respect to x and y respectively we get z a p a x
(((((( "!#
z bqa y
((((((. "+#
ubstitute "!# and "+# in "&# we get z px qy p ! q ! , which is the required partial differential equation.
&+' Form t(e partial differential eqation "$ eliminatin) t(e ar"itrar$ %on#tant#
a, ", % from
x ! a!
y ! b!
z ! c!
&*
Soltion!
e note that the number of constants is more than the number of independent variable. Hence the order of the resulting equation will be more than &. x ! a!
y ! b!
z ! c!
&
.................. "&#
*ifferentiating *ifferentiating "&# partially with respect to x and then with respect to y, we get
! x !
a ! y b!
! y c! ! z c!
p -
....................."!#
q-
......................"+#
*ifferentiating *ifferentiating "!# partially with respect to x, & a
!
& c
!
" zr p ! #
(((((.."#
! z here r . x ! /rom "!# and "# ,
c! a
!
c! a!
zp
.......... .......... .."1#
x
zr p
. !
.......... .......... .."0#
/rom "1# and "0#, we get
!
! ! 'iven z ax by a b
(((((... "&#
Here we have two arbitrary constants a ) b. *ifferentiating equation "&# partially with respect to x and y respectively we get z a p a x
(((((( "!#
z bqa y
((((((. "+#
ubstitute "!# and "+# in "&# we get z px qy p ! q ! , which is the required partial differential equation.
&+' Form t(e partial differential eqation "$ eliminatin) t(e ar"itrar$ %on#tant#
a, ", % from
x ! a!
y ! b!
z ! c!
&*
Soltion!
e note that the number of constants is more than the number of independent variable. Hence the order of the resulting equation will be more than &. x ! a!
y ! b!
z ! c!
&
.................. "&#
*ifferentiating *ifferentiating "&# partially with respect to x and then with respect to y, we get
! x !
a ! y b!
! y c! ! z c!
p -
....................."!#
q-
......................"+#
*ifferentiating *ifferentiating "!# partially with respect to x, & a
!
& c
!
" zr p ! #
(((((.."#
! z here r . x ! /rom "!# and "# ,
c! a
!
c! a!
zp
.......... .......... .."1#
x
zr p
. !
.......... .......... .."0#
/rom "1# and "0#, we get
!
!
! z z z xz ! x z , which is the required partial differential x x x
equation.
&-' Find t(e differential eqation of all #p(ere# of t(e #ame radi# % (a.in) t(eir %enter on t(e $o/0plane*
*
Soltion!
The equation of a sphere having its centre at -, a, b , that lies on the yoz 2 plane and having its radius equal to c is x " y a # " z b# c !
!
!
!
(((((. "&#
3f a and b are treated as arbitrary constants, "&# represents the family of spheres having the given property. property. *ifferentiating "&# partially with respect to x and then with respect to y, we have
and
! x ! z b p -
((((( "!#
! y a ! z b q -
(((((."+#
z b
/rom "!#,
x p
4sing "# in "+#, y b
(((((."#
qx p
(((((.."1#
4sing "# and "1# in "&#, we get x !
. i.e.
& p
!
q ! x ! p
!
x ! p
!
c!
q ! x ! c ! p ! , which is the required partial differential equation.
Pro"lem#
Formation of partial differential eqation "$ elimination of ar"itrar$ fn%tion#!
+
&1'Form t(e partial differential eqation "$ eliminatin) t(e ar"itrar$ fn%tion f2 from z e f x by ay
ay #oltion% 'iven z e f x by
e ay z f x by
i.e.
((((("&#
*ifferentiating *ifferentiating "&# partially with respect to x and then with respect to y, we get e ay p f 5 u . &
((((."!#
e ay q ae ay z f 5 u .b
(((((."+#
where u x by
6liminating f7"u# from "!# and "+#, we get q az p
b
q az bp
i.e.
&+' Form t(e partial differential eqation "$ eliminatin) t(e ar"itrar$ fn%tion 2
z ! xy ,
x
-
z
Soltion! 'iven
8et
z ! xy ,
x
-
(((((("&#
z
v
u z ! xy ,
x z
Then the given equation is of the form u , v - . The elimination of from equation "!#, we get,
u x u y ! zp y
i.e.
! zq x
v x v y
z px z ! xq z !
i.e
i.e
xq z px ! ! zq x ! z z
! zp y
px ! q xy ! z ! zx
&-' Form t(e partial differential eqation "$ eliminatin) t(e ar"itrar$ fn%tion f2 from z f ! x y g + x y
Soltion! 'iven z f ! x y g + x y
.(((("&#
*ifferentiating "&# partially with respect to x, p f u .! g v . +
((((."!#
here u ! x y and v + x y
*ifferentiating "&# partially with respect to y, q f u .& g v " &#
((((. "+#
*ifferentiating "!# partially with respect to x and then with respect to y, r f u . g v . 9
((((. "#
and s f u .! g v ." +#
((((.. "1#
*ifferentiating "+# partially with respect to y, t f u .& g v . &
((((.. "0#
6liminating f u and g v from "#, "1# and "0# using determinants, we
have
.
9
r
!
+
s
&
&
t
:-
i.e.
1r 1 s +-t -
or
! z ! z ! z 0 x ! x y y !
&3' Form t(e partial differential eqation "$ eliminatin) t(e ar"itrar$ fn%tion 2 from z
& x
y x y x *
Soltion! 'iven
z
& x
u u
((((..."&#
here u y x 1
*ifferentiating partially with respect to x and y, we get p q
r
& x & x
& x
s t
& x
u "&#
& x
!
u u "&#
u .& u . &
u .& &
x
! x
u
!
u !
& x
(((("!# ((((."+#
! x
+
u u . &
u u "&#
u .& u . &
(((.."# (((("1# ((((.."0#
/rom "# and "0#, we get r t
: :
!
!
x
x +
u !
u
! & u u ! x x
! x !
z
! z ! z ! ! z ! x y
x !
i.e.
Soltion# of partial differential eqation# ;onsider the following two equations z ax by
and
y x
z xf
(((.."&# (((.."!#
6quation "&# contains arbitrary constants a and b, but equation "!# contains only one arbitrary function f. 3f we eliminate the arbitrary constants a and b from "&# we get a partial differential equation of the form xp yq z . 3f we eliminate the arbitrary function f from "!# we get a partial differential equation of the form xp yq z . Therefore for a given partial differential equation we may have more than one type of solutions.
0
T$pe# of #oltion#!
"a# A solution in which the number of arbitrary constants is equal to the number of independent variables is called Complete Inte)ral &or' Complete #oltion* "b# 3n complete integral if we give particular values to the arbitrary constants we get Parti%lar Inte)ral*
"c# The equation which does not have any arbitrary constants is known as Sin)lar Inte)ral*
To find t(e )eneral inte)ral! uppose that f x, y , z , p, q -
(((...."&#
is a first order partial differential equation whose complete solution is x , y , z , a , b -
(((.."!#
here a and b are arbitrary constants. 8et b f a , where
(((."+#
*ifferentiating "+# partially with respect to
. f a - a b
(((."#
The eliminant of
4et(od# to #ol.e t(e fir#t order partial differential eqation! T$pe 1! Eqation of t(e form f " p, q# -
(((..."&#
i.e the equation contains p and q only. uppose that
z ax by c
((....."!#
is a solution of the equation
=
z z a, b x y p a, q b
substitute the above in "&#, we get f "a, b# -
on solving this we can get b a , where is a known function. 4sing this value of b in "!#, the complete solution of the given partial differential equation is z ax a y c
(((("+#
is a complete solution, To find the singular solution, we have to eliminate
*ifferentiating the above with respect to
and -:&. The last equation is absurd. Hence there is no singular solution for the equation o f Type &.
Pro"lem#!
! ! &1' Sol.e p q & *
Soltion!
! ! 'iven% p q &
((((."&#
6quation "&# is of the form f " p, q# - .
Assume z ax by c
((((."!#
be the solution of equation "&#. /rom "!# we get p a, q b .
"&# a ! b ! & b & a!
(((."+#
ubstitute "+# in "!# we get z ax
& a ! y c
((...."#
>
This is a complete solution.
To find the general solution% e put c f a in "#,where
i.e. z ax & a ! y f a
(((("1#
*ifferentiating "1# partially with respect to
x
a & a!
y f a -
((((("0#
6liminating
To find the singular solution% *ifferentiate "# partially with respect to
- x
& a!
,
-:&."which is absurd# so there is no singular solution. &+' Sol.e p q pq Soltion!
'iven% p q pq
(((.."&#
6quation "&# is of the form f " p, q#
Assume z ax by c
((("!#
be the solution of equation "&#. /rom "!# we get p a, q b
"&# a b ab
b
a a &
((....."+#
ubstituting "+# in "!#, we get z ax
a a &
y c
((("#
This is a complete solution.
9
To find the general solution% e put c f a in "#, we get z ax
a a &
y f a
((.."1#
*ifferentiating "1# partially with respect to
x
&
a & !
y f a
(((.."0#
6liminating
To find the singular solution% *ifferentiating "# with respect to
&
a & !
y,
and -:& "which is absurd#. o there is no singular solution.
T$pe +! &Clairat2# t$pe' T(e eqation of t(e form z px qy f " p, q#
((("&#
i# 5no6n a# Clairat2# eqation .
Assume
z ax by c
(((("!#
be a solution of "&#. z z a, b x y p a, q b
ubstitute the above in "&#, we get z ax by f "a, b#
(((.."+#
which is the complete solution.
&-
Pro"lem!
z px qy
&1' Sol.e
pq
Soltion!
'iven%
z px qy
pq
(((."&#
6quation "&# is a ;lairaut7s equation
8et z ax by c
(((.."!#
be the solution of "&#.
?ut p a, q b in "&#, we get z ax by
ab
(((."+#
which is a complete solution.
To find the general solution% e put b f a in "+#, we get z ax f a y
af a
((("#
*ifferentiate "# partially with respect to
& ! af a
a f a f a
(((.."1#
6liminating
To find singular solution, *ifferentiate "+# partially with respect to
& ! ab
x
b ! a
.b
(((.."0#
*ifferentiate "+# partially with respect to
& ! ab
y
a ! b
.a
(((.."=#
@ultiplying equation "0# and "=#,we get
&&
b a ! a ! b
xy
xy xy
&+' Sol.e
&
&
z px qy
q
p
p
Soltion!
'iven% z px qy
q
p
p
(((."&#
6quation "&# is a ;lairaut7s equation
8et
z ax by c
(((..."!#
be the solution of "&#.
?ut p a, q b in "&#, we get z ax by
b a
a
(((."+#
which is the complete solution.
To find the general solution% e put b f a in "+#, we get z ax f a y
f a a
a
((.."#
*ifferentiate "# partially with respect to
a f a f a a
&
((.."1#
6liminating
To find the singular solution% *ifferentiate "+# partially with respect to
x
b a
!
b a
!
&
&
&!
a ! x b &
b a!x &
.............. "#
*ifferentiate "+# partially with respect to
a
& y
& a
((((."1#
ubstituting equation "# and "1# in equation "+#, we get
& a ! x & & ! z x a x & y & y y y z z
x y
x y
a ! yx y a ! xy y
& y
& y
yz & x
T$pe -! Eqation# not %ontainin) 7 and $ e7pli%itl$, i*e* eqation# of t(e form f z , p, q -
(((."&#
/or equations of this type ,it is known that a solution will be of the form z x ay
(((."!#
here
and
q
p
dz u dz . du x du
dz u dz a . du y du
3f "!# is to be a solution of "&#, the values of p and q obtained should satisfy "&#.
dz dz ,a - i.e. f z , du du
((.."+#
/rom "+#, we get
&+
dz du
z , a
(((."#
ow "# is a ordinary differential equation, which can be solved by variable separable method. The solution of "#, which will be of the form g z , a u b or g z , a x ay b , is the complete solution of "&#. The general and singular solution of "&# can be found out by usual method.
Pro"lem#!
! ! &1'Sol.e z p q .
Soltion! ! ! 'iven% z p q
(((("&#
6quation "&# is of the form f z , p, q
Assume z u where, u x ay be a solution of "&#. p
z dz u dz . x du x du
q
z dz u dz . a y du y du
p
dz du
qa
dz du
((."!#
(("+#
ubstituting equation "!# ) "+# in "&#, we get
!
dz ! dz z a du du
!
!
dz ! & a du
z
!
z dz ! & a du &
dz du
z !
& a !
$y variable separable method, dz & !
z
du
& a !
& !
&
$y integrating, we get
dz &
&
& ! +
& a
z !
! z
x ay
& a !
z
c
& !
x ay
! & a z
du c
x ay
!
! & a
&
& ! !
k
c !
(((."#
!
This is the complete solution.
To find the general solution% e put k f a in "#, we get
z
x ay
! & a
& ! !
f
a
((.."1#
*ifferentiate "1# partially with respect to
y xa !& a
!
+
f a
(((.."0#
!
6liminating
To find the singular solution% *ifferentiate "# partially with respect to
-
y xa !& a
and
+ ! !
(((.."=#
- & "which is absurd#
o there is no singular solution.
&1
! ! &+'Sol.e 9 p z q *
Soltion!
! ! 'iven% 9 p z q
((("&#
6quation "&# is of the form f z , p, q
Assume z u where , u x ay be a solution of "&#. p
z dz u dz . x du x du
q
z dz u dz . a y du y du
p
dz du
qa
dz du
((."!#
(("+#
ubstituting equation "!# ) "+# in "&#, we get ! dz ! ! dz 9 z a du du
!
dz ! 9 z a du !
dz 9 z a ! du dz du
!
&
+
z a
.
z a ! dz
! +
!
du
3ntegrating the above, we get
z a !
&
dz
!
+
! z a ! +
! z a ! +
!
+
du c
! uc +
! x ay c +
+ !
z a
!
+ ! !
x ay k
(((.."#
This is the complete solution.
To find the general solution%
&0
e put k f a in "#, we get
z a
+ ! !
x ay f
a
((.."1#
*ifferentiate "1# partially with respect to
+a z a
!
& !
y f a
((.."0#
6liminating
+a z a
and
& ! !
y
(((.."=#
- & "which is absurd#
o there is no singular solution.
T$pe 3! Eqation# of t(e form f x, p g y , q
(((.."&#
i.e. equation which do not contain z explicitly and in which terms containing p and x can be separated from those containing q and y. To find the complete solution of "&#, e assume that f x, p g y, q a .where
ow dz
z z dx dy or pdx qdy x y
i.e. dz x, a dx y, a dy 3ntegrating with respect to the concerned variables, we get z
x, a dx y, a dy b
(((."!#
The complete solution of "&# is given by "!#, which contains two arbitrary constants
&=
Pro"lem#! &1'Sol.e pq xy . Soltion!
'iven% pq xy
p x
y q
(((.."&#
6quation "&# is of the form f x, p g y , q
p
8et
x
p x
y
a "say#
q
p ax
a
y a q
q
y
imilarly,
Assume dz pdx qdy be a solution of "&#
a
((((."!#
(((("+#
ubstitute equation "!# and "+# to the above, we get y
dz axdx
a
dy
3ntegrating the above we get, &
dz a xdx a ydy c z
ax
!
!
y
! z ax !
!
!a y ! a
c k
(((.."#
This is the complete solution. The general and singular solution of "&# can be found out by usual method. &+' Sol.e p q x y * Soltion!
'iven% p q x y
p x q y
((((.. "&#
6quation "&# is of the form f x, p g y, q
8et
p x y q a "say#
&>
p x a
p x a
((((. "!#
q y a
((((("+#
imilarly, y q a
Assume dz pdx qdy be a solution of "&# ubstitute equation "!# and "+# to the above, we get dz x a dx y a dy
3ntegrating the above we get,
dz x a dx y a dy c x y a x y c z !
!
!
! ! ! z x y !a x y c
((((("#
This is the complete solution. The general and singular solution of "&# can be found out by usual method.
Eqation# red%i"le to #tandard t$pe#0tran#formation#! T$pe A! m n m n Eqation# of t(e form f x p, y q - or f x p, y q, z - *
here m and n are constants, each not equal to &. m
e make the transformations x & p
Then
q
X and y & n Y .
z z X & m x m P , . x X x
z z Y . & n y n Q, y Y y
where P
where Q
z and X
z Y
m n Therefore the equation f x p, y q - reduces to f & m P , & n Q -, .which is a
type & equation. The equation f x m p, y n q, z - reduces to f & m P , & n Q, z -, .which is a type + equation.
Pro"lem! ! ! ! &1'Sol.e p x y zq ! z .
Soltion!
! ! ! 'iven% p x y zq ! z
&9
This can be written as
px
! !
qy ! z ! z ! .
hich is of the form f x m p , y n q, z - , where m:!,n:!.
& m ?ut X x
& x
B Y y & n
& y
P
z z x . p x ! px ! X x X
Q
z z y . q y ! qy ! Y y Y
ubstituting in the given equation, P ! qz ! z ! .
This is of the form f p , q, z - . 8et Z f X aY , where u X aY P
dz du
,Qa
6quation becomes,
dz du
. !
dz dz ! z ! az du du
olving for
dz du
,
dz du dz z
log z
log z
az
a ! z ! > z ! !
a
a! > !
a
a! > !
a
du
X aY b
a > & !
!
a b is a complete solution. x y
The general and singular solution can be found out by usual method.
T$pe 8! k k k k Eqation# of t(e form f z p, z q - or f z p, z q, x, y - *
here k is a constant, which is not equal to 2&. e make the transformations Z z k & . Then
P
Z k & z k p and x
!-
Q
Z k & z k p y
Q P , -, which is a type & Therefore the equation f z k p, z k q - reduces to f k & k &
equation. The equation
Q P , , x, y -, which is a k & k &
f z k p , z k q, x, y - reduces to f
type equation.
Pro"lem#! ! ! &1'Sol.e! z q z p &.
Soltion!
! ! 'iven% z q z p &.
The equation can be rewritten as z ! q z ! p & !
((("&#
hich contains z ! p and z ! q . Hence we make the transformation Z z + P
Z + z ! p x
z ! p
! imilarly z q
P +
Q +
4sing these values in "&#, we get Q ! +P 9
(((.."!#
As "!# is an equation containing ? and C only, a solution of "!# will be of the form Z ax by c
((((."+#
ow P a and Q b, obtained from "+# satisfy equation "!# b ! +a 9
i.e. b +a 9
Therefore the complete solution of "!# is Z ax
+a
9 y c
i.e complete solution of "&# is z + ax +a 9 y c ingular solution does not exist. 'eneral solution is found out as usual. T$pe C!
!&
m k n k Eqation# of t(e form f x z p, y z q - , 6(ere m, n &B k &
e make the transformations X x & m , Y y & n and Z z k &
Z dZ z dx . . X dz x dX
P
Then
k & z k p.
Q k & z q.
& m
y n
k
and
x m
& n
Therefore the given equation reduces to
& m & n P , Q k & k &
f
This is of type & equation.
Pro"lem!
p ! q ! &1'Sol.e z ! ! & y x !
Soltion!
p ! q ! 'iven% z ! ! & y x !
3t can be rewritten as
x zp &
!
y & zq & !
(((.."&#
which is of the form x m z k p y n z k q & !
!
we make the transformations X x & m , Y y & n and Z z k &
i.e. Then
p
X x ! , Y y ! and Z z !
z dz Z dX & . . . P .! x x dZ X dx ! z
P x & zp ,
imilarly,
Q y & zq ,
4sing these in "&#,it becomes P ! Q ! &
(((("!#
As "!# contains only ? and C explicitly, a solution of the equation will be of the
!!
form Z aX bY c
((((."+#
Therefore P a and Q b, obtained from "+# satisfy equation "!#
i.e.
a ! b ! &,
b & a!
Therefore the complete solution of "!# is Z aX & a ! Y c
Therefore the complete solution of "&# is z ! ax !
& a ! y ! c
ingular solution does not exist. 'eneral solution is found out as usual.
T$pe D!
px qy , Eqation of t(e form f z z $y putting X log x, Y log y and Z log z the equation reduces to f P , Q -,
Z Z where P X and Q Y .
Pro"lem#! &1'Sol.e pqxy z * !
Soltion!
'iven% pqxy z ! .
(((((."&#
Dewriting "&#,
px qy & . z z
(((((."!#
px qy and , we make the substitutions As "!# contains z z X log x, Y log y and Z log z
Then
i.e.
P
px z
z dz Z dX & z . P . . . x dZ X dx x P
!+
imilarly,
qx z
Q
4sing these in "!#, it becomes PQ &
((((.."+#
which contains only ? and C explicitly. A solution of "+# is of the form Z aX bY c
(((("#
Therefore P a and Q b, obtained from "# satisfy equation "+# i.e.
ab & or b
& a &
Therefore the complete solution of "+# is Z aX Y c a
Therefore the complete solution of "&# is log z a log x
& a
log y c ((.."1#
'eneral solution of "&# is obtained as usual.
9eneral #oltion of partial differential eqation#! ?artial differential equations, for which the general solution can be obtained directly, can be divided in to the following three categories. "&# 6quations that can be solved by direct "partial# integration. "!# 8agrange7s linear equation of the first order. "+# 8inear partial differential equations of higher order with constant coefficients.
Eqation# t(at %an "e #ol.ed "$ dire%t &partial' inte)ration % Pro"lem#! &1'Sol.e t(e eqation
u !u - 6(en x -. e t cos x, if u - 6(en t - and t xt
Al#o #(o6 t(at u sin x , 6(en t ! * Soltion!
'iven%
!u e t cos x, xt
(((((."&#
3ntegrating "&# partially with respect to x,
u e t sin x f t t
(((((."!#
!
hen t - and
u - in "!#, we get f t - . t
6quation "!# becomes
u e t sin x t
(((((."+#
3ntegrating "+# partially with respect to t, we get u e t sin x g x
((((("#
4sing the given condition, namely u - when t - , we get - sin x g x or g x sin x
4sing this value in "#, the required particular solution of "&# is u sin x & e t
ow
lim u sin x lim & e t !
t !
t
sin x i.e. when t !, u sin x .
&+' Sol.e t(e eqation
z z + x y and x cos y #imltaneo#l$* x y
Soltion! 'iven z x
+ x y
z x cos y y
................"&# .......... ........" !#
3ntegrating "&# partially with respect to x, z
+ x
!
!
yx f y
((((.."+#
*ifferentiating "&# partially with respect to y, z x f y y
((((..."#
;omparing "!# and "#, we get f y cos y f y sin y c
..................." 1#
Therefore the required solution is z
+ x ! !
yx sin y c , where c is an arbitrary constant.
!1
La)ran)e2# linear eqation of t(e fir#t order! A linear partial differential equation of the first order , which is of the form Pp Qq R
where P , Q , R are functions of x, y , z is called 8agrange7s linear equation.
6or5in) rle to #ol.e
Pp Qq R
"&#To solve Pp Qq R , we form the corresponding subsidiary simultaneous equations dx P
dy Q
dz R
.
"!#olving these equations, we get two independent solutions u a and v b . "+#Then the required general solution is f u , v - or u v or v u .
Soltion of t(e #imltaneo# eqation#
dx P
dy Q
dz R
.
4et(od# of )ropin)!
$y grouping any two of three ratios, it may be possible to get an ordinary differential equation containing only two variables, eventhough ? BCBD are in general, functions of x,y,z. $y solving this equation, we can g et a solution of the simultaneous equations. $y this method, we may be able to get two independent solutions, by using different groupings.
4et(od# of mltiplier#!
3f we can find a set of three quantities l,m,n which may be constants or functions of the variables x,y,z, such that
lP mQ nR - ,
then the solution of the
simultaneous equation is found out as follows. dx P
ince
dy Q
dz R
ldx mdy ndz lP mQ nR
lP mQ nR -, ldx mdy ndz -. 3f
ldx mdy ndz - is
an exact
differential of some function u x, y, z , then we get du -. 3ntegrating this, we get
u a , which is a solution of
dx P
dy Q
dz R
.
imilarly, if we can find another set of independent multipliers l , m , n , we can get another independent solution v b .
!0
Pro"lem#! &1'Sol.e xp yq x * Soltion!
'iven% xp yq x . This is of 8agrange7s type of ?*6 where P x, Q y , R x * The subsidiary equations are Taking first two members
dx x
dy
y
dz x
.
dy
y
x c& y
i.e.
u
Taking first and last members
x
log x log y log c&
3ntegrating we get
dx
dx x
x y
dz x
((((."&#
.
dx dz .
i.e.
x z c !
3ntegrating we get
v x z
(((......"!# x
i.e.
Therefore the solution of the given ?*6 is u , v -
y
, x z - .
&+'Sol.e t(e eqation x ! z p ! z y q y x. Soltion!
'iven% x ! z p ! z y q y x. This is of 8agrange7s type of ?*6 where P x ! z , Q ! z y , R y x * The subsidiary equations are
dx x ! z
dy ! z y
4sing the multipliers &,&,&, each ratio in "&#:
dz y x
.
(((."&#
dx dy dz -
*
dx dy dz - *
3ntegrating, we get
x y z a
4sing the multipliers y,x,!z, each ratio in "&#:
((((("!' ydx xdy ! zdz -
*
d xy ! zdz - *
!=
xy z ! b
3ntegrating, we get
((((("+#
! Therefore the general solution of the given equation is f x y z , xy z - .
! ! ! ! &-'S(o6 t(at t(e inte)ral #rfa%e of t(e PDE x y z p y x z q x y z * ! ! :(i%( %ontain# t(e #trai)(t line x y -, z & is x y ! xyz ! z ! - *
Soltion!
The subsidiary equations of the given 8agrange
x y z !
4sing the multipliers
&
,
&
,
dy
y x z
&
x y z & x
dx
!
dz
x
!
.
y ! z
& , * each
ratio in "&#: x
dx
& y
dy
& z
dz
*
& y
dy
& z
dz - *
xyz a
3ntegrating, we get
((((("&#
4sing the multipliers y,x,2&, each ratio in "&#:
((((("!' xdx ydy dz -
*
xdx ydy dz - * ! ! 3ntegrating, we get x y ! z b
((((("+#
The required surface has to pass through x y z &
((((("#
4sing "# in "!# and "+#, we have x ! a ! x ! ! b
((((("1#
6liminating x in "1# we get, !a b ! -
(((((."0#
ubstituting for a and b from "!# and "+# in "0#, we get x y ! xyz ! z ! - , which is the equation of the required !
!
surface.
Linear P*D*E*S of (i)(er order 6it( %on#tant %oeffi%ient# % The standard form of a homogeneous linear partial differential equation of the
!>
n th order with constant coefficients is
n z n z n z n z a& n & a ! ... a n R x, y a x n x y x n ! y ! y n
(((((."&#
where a7s are constants. 3f we use the operators
x
and
y , we can symbolically write equation "&#
as
a a . . an z R x y, n n& - &
n
(((((."!#
f , z R x, y
(((((."+#
where f , is a homogeneous polynomial of the n th degree in and . The method of solving "+# is similar to that of solving ordinary linear differential equations with constant coefficients. The general solution of "+# is of the form z : "complementary function#E"particular integral#,where the complementary function is the D.H. of the general solution of f , z - and the particular integral is given symbolically by
& f ,
R x, y .
Complementar$ fn%tion of f , z R x, y %
;./ of the solution of f , z R x, y is the D.H. of the solution of f , z - .
(((("&#
3n this equation, we put m, & ,then we get an equation which is called the a7iliar$ eqation .
Hence the auxiliary equation of "&# is a- m n a& m n & ... a n -
8et the roots of this equation be
((((."!#
m& , m ! ,...m n .
Ca#e 1!
The roots of "!# are real and distinct. The general solution is given by
z & y m& x ! y m! x ... n y m n x
!9
Ca#e +!
Two of the roots of "!# are equal and others are distinct. The general solution is given by z & y m& x ! y m& x ... n y mn x
Ca#e -!
z & y m& x x ! y m& x ... x r & r y m r x To find parti%lar inte)ral! Rle &1'% 3f the D.H. of a given ?*6 is f x, y e ax by , then
P . !
& f ,
e ax by
?ut a, b P . !
& f a, b
e ax by
if f a, b -
3f f a, b -, refer to Dule "#. Rle &+'% 3f the D.H. of a given ?*6 is f x, y sin ax by or cos ax by , then P . !
& f , !
Deplace a
!
sin ax by or cos ax by
,
!
b
!
and ab in f , provided the
denominator is not equal to zero. 3f the denominator is zero, refer to Dule "#. m n Rle &-'% 3f the D.H. of a given ?*6 is f x, y x y , then
P . !
& f ,
x m y n
f , & x m , y n
6xpand f ,
&
by using $inomial Theorem and then operate on x m y n .
Rle &3'% 3f the D.H. of a given ?*6 f x, y is any other function Fother than Dule"&#,
"!# and"+#G resolve f , into linear factors say m& m! etc. then the
+-
P . !
&
m& m!
f x, y
Note! 3f the denominator is zero in Dule "&# and "!# then apply Dule "#. :or5in) rle to find P*I 6(en denominator i# /ero in Rle &1' and Rle &+'*
3f the D.H. of a given ?*6 is of the form sin ax by or cos ax by or e ax by P . !
Then
&
b a
. f ax by
x n
f ax by b n nH
This rule can be applied only for equal roots.
Pro"lem#!
+ ! + x! y x y
+ ! z e e
&1' Sol.e
Soltion!
+ ! + x! y x y
+ ! z e e
'iven%
The auxiliary equation is m + +m ! m &,&,! # . " xf & y x f ! y x f + y ! x
P . !
&
+
!
!
+ !
&
!
+
&
& ! x y x ! x y e xe 9 !
9 !
! x y
! x y e !
.
&
e
e ! x y
&
.
x y
e
&
! ! &
9
!
e x y
e x y
The general solution of the given equation is
+&
z xf & y x f ! y x f + y ! x
&+'Sol.e
x 9
e ! x y
x ! &>
e x y
. 1 z x y sin x! y+ !
!
Soltion!
. 1 z x y sin x! y+
'iven%
!
!
The auxiliary equation is m ! m 1 m &,1
# . " & y 1 x ! y x
P . ! &
& ! 1 ! &
xy
& ! 1
xy
!
&
& ! & ! 1 xy
& & 1 ... xy ! ! &
&
xy !
+
x y
. xy
&
x 0 & & x + y x 1 0 +
P ! . !
& !
1
!
sin ! x + y
+!
&
! . !.+ 1 + ! !
& &=
sin ! x + y
sin ! x + y
Therefore the general solution is z & y 1 x ! y x
0
x y
&
+
+-
x 1
& &=
sin ! x + y
! ! ! x y
!
&-'Sol.e
&
! z x y e
Soltion!
!
! ! ! x y
! z x y e
'iven%
The auxiliary equation is m ! !m & m &,&
# . " xf & y x f ! y x P . !
&
e x y e x y
!
e x y x ! y ! &
& & &
!
!
x y
!
x ! y !
!
++
!
e & x y x y & ! e & + x y &
x y
! !
!
!
! !
!
e x y
!
& ! ! ! + ! x y ! x y ! x ! ! !
& & & e x y y ! ! x ! y. + x ! 0. x ! & & & x y ! x 1 y x 0 e x y &1 0- &!
Therefore the general solution is & ! x y & ! & y xy x x e &1 0- &!
z xf & y x f ! y x
+
43T & ?art A "&#/orm a partial differential equation by eliminating arbitrary constants a and b from !
z x a y b
!
An#! ! ! 'iven z x a y b p q
z
! x a
!
! y b
+
x z y
((. "&#
ubstituting "!# ) "+# in "&#, we get z "!# olve% ! z !
p !
q!
!
An#!
Auxiliary equation m ! !m & -
m & !
-
m &,& z f & y x xf ! y x
"+#/orm a partial differential equation by eliminating the arbitrary constants a and b from ! ! the equation x a y b z ! cot ! " . An#! ! ! 'iven% x a y b z ! cot ! " (.. "&# ?artially differentiating with respect to
+1
z ! p ! cot " z ! q ! cot " z ! cot ! " .
p ! q !
tan ! " .
! ! "#/ind the complete solution of the partial differential equation p q pq -.
An#!
'iven% p ! q ! pq -.
((.. "&#
8et us assume that
z ax by c
((( "!#
be the solution of "&# ?artially differentiating with respect to
a b b a!
(((. "#
ubstituting "1# in "!# we get z b & a ! x by c
"1#/ind the ?*6 of all planes having equal intercepts on the x and y axis. An#! The equation of such plane is x a
y a
z b
&
(((. "&#
?artially differentiating "&# with respect to
p
q
b b
- p
b
-q
b
a a
(((.. "!# (((.. "+#
/rom "!# and "+#, we get p q "0#/ind the solution of px ! qy ! z ! . An#! The .6 is dx x !
dy y !
dz z !
Taking first two members, we get
+0
dx x
!
dy y !
3ntegrating we get & & c& x
y
&
u
i.e
y
&
c&
x
Taking last two members, we get dy y
!
dz z !
3ntegrating we get & & c! y
z
&
v
i.e
z
&
c!
y
The complete solution is & & & & , y x z y ! ! "=#/ind the singular integral of the partial differential equation z px qy p q . An#!
The complete integral is z ax by a ! b ! . z a z b
x !a
-a
y !b - b
x !
y !
Therefore z
x ! !
y !
!
x !
y !
x !
y !
y ! x ! z
">#olve%
p ! q ! m ! .
An#! $iven
p ! q ! m !
((.. "&#
8et us assume that
z ax by c
(( "!#
be the solution of "&# ?artially differentiating with respect to
+=
z a x z q b y ubstituting "+# in "&# we get p
((.. "+#
a! b! m!
This is the required solution. "9#/orm a partial differential equation by eliminating the arbitrary constants a and b from z ax n by n . An#! $iven z ax n by n .
((( "&# ?artially differentiating with respect to
z
& n
p nx
n &
x n
q ny
n &
yn
px qy
This is the required ?*6. "&-#olve%
+ !
! +
z .An#!
Auxiliary equation m+ m! m & -
m & + m &,&,& z f & y x xf ! y x x ! f + y x
"&&#/orm a partial differential equation by eliminating the arbitrary constants a and b from z x ! a ! y ! b ! . An#%
! ! ! ! 'iven z x a y b
((. "&#
+>
p z ! x y ! b ! y ! b ! ! x x q z q ! y x ! a ! x ! a ! ! y y
p
ubstituting "!# ) "+# in "&#, we get z
! + q p . ! y ! x
pq xyz
"&!#olve%
!
& z -
An#!
The given equation can be written as & & z e know that the ;./ corresponding to the factors
m& " & m! " ! z z e x f & y m& x e x f ! y m ! x " &
is
" !
3n our problem " & &, " ! &, m& -, m! &
# . " e x f & y x e x f ! y x z e x f & y x e x f ! y x
"&+#/orm a partial differential equation by eliminate the arbitrary function f from xy z f . z An#! xy . z
$iven % z f
xy zy xy. p .. z ! z xy zx xy.q q f .. z ! z p f
/rom "&#, we get ! xy pz f z zy xyp
.......... .."&#
............"!#
............."+#
ubstituting "+# in"!#, we get pz ! zy xy. p . p zy xyp z ! "&#olve%
+9
+ !
!
+
! ! z .An#% Auxiliary equation m + !m ! m ! m &,&,! %olution is z f & y x f ! y x f + y ! x
"&1#Ibtain partial differential equation by eliminating arbitrary constants a and b from
x a ! y b ! z ! &. An#! ! ! 'iven x a y b z ! & ! x a ! zp - x a zp ! y b ! zq - y b zq ubstituting "!# ) "+# in "&#, we get z ! p ! z ! q ! z ! & p ! q ! &
((. "&# ........."!#
+
& z !
"&0#/ind the general solution of ! z ! z ! z &! 9 ! -. x y y ! y An#!
Auxiliary equation is m ! &!m 9 m
m
&!
& & >
+ + , ! !
'eneral solution is + + z f & y x xf ! y x ! !
"&=#/ind the complete integral of p q pq ,
where
p
z z ,q . x y
An#% 8et us assume that
z ax by c
((( "&#
be the solution of the given equation. ?artially differentiating with respect to
-
z a x z q b y ubstituting "!# in "&# we get p
a b ab b
((.. "!#
a a &
ubstituting the above in "&# we get a z ax y c a & This gives the complete integral. "&>#olve%
+
! +
+ ! z An#!
Auxiliary equation m + +m ! m &,&,! %olution is z f & y x xf ! y x f + y ! x "&9#/ind the ?*6 of the family of spheres having their centers on the line x:y:z. An#% The equation of such sphere is
x a ! y a ! z a ! r ! ?artially differentiating with respect to
..........."!#
/rom "&#, a
x zp & p
..........."+#
/rom "!#, a
y zq & q
..........."#
/rom "+# and "#, we get x zp y zq & p & q
This is the required ?*6. "!-#olve%
+ z ! z + z + z -. ! > x + x ! y x y ! y + &
An#%
Auxiliary equation m + !m ! m > m !,!,! %olution is z f & y ! x xf ! y ! x f + y ! x
?art $ "&#"i# /orm a partial differential equation by eliminating arbitrary functions from z xf ! x y g ! x y "ii# olve% p ! y & x ! qx ! ! ! ! ! ! ! "!#"i# olve% x z y p y x z q z y x + z + z ! e x ! y sin x y "ii# olve% + ! x x y
! ! ! "+#"i# olve% x y z p ! xyq ! xz -
!
!
x+ y.
! z x! y+ e
"ii# olve%
"#"i# olve% z ! p ! q ! x ! y !
!
!
+ z s yin
"ii# olve%
! ! ! ! "1#"i# olve% x y yz p x y xz q z x y
"ii# olve%
! !- ! z e1 x y sin . x y
! ! "0#"i# olve% z p q
!
! ! x+ y
0 z x ye
"ii# olve%
! ! "=#"i# olve% y z p xyq xz -
!
!
! x
0 1 z x ye si ynh
"ii# olve%
">#"i# olve% p & q ! q & z
!
x! y
!
. . z x y e
"ii# olve%
! ! ! "9#"i# olve% x yz p y zx q z xy "ii# olve% x y z p y z x q z x y
! ! ! ! ! ! "&-#"i# olve% p x y q x z
"ii#olve%
+ + z xy !
!
"&&#"i# /orm the partial differential equation by eliminating f and from z f y x y z
! z x y e !
"ii# olve%
+
! x
"&!#"i# /ind the complete integral of p q x y ! "ii# olve% y p xyq x z ! y "&+#"i# olve% + z y p x ! z q ! y + x
"ii# olve%
!
e
!
! + + ! z
+ x
!e
! y
!
. 1 z s xin y! e !
x! y
!
"iii# olve%
"&#"i# olve z & p q "ii# olve% y z p ! x y q ! x z !
!
!
"&1#"i# /orm a partial differential equation by eliminating arbitrary functions f and g in z x ! f y y ! g x
+
