UNIT I PARTIAL DIFFERENTIAL EQUATIONS MA6351 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS by B.Saravanan Assistant Professor Department of Applied Mathematics SVCE 1
UNIT I PARTIAL DIFFERENTIAL EQUATIONS
MA6351 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
byB.Saravanan
Assistant ProfessorDepartment of Applied Mathematics
SVCE
1
�Formation of partial differential equations
� Lagrange’s linear equation
� Solutions of standard types of first order partial differential equations
� Linear partial differential equations of second and higher order with constant coefficients
Syllabus
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Partial Differential Equation
Partial differential equation is one which involves partial derivatives. The order of PDE is the order of highestderivative occurring in it.
.,,,,
:
.var
,var,),(
2
22
2
2
y
zt
yx
zs
x
zr
y
zq
x
zp
Notation
iabledependentisz
iabletindependenareyxwhereyxfzIf
∂∂=
∂∂∂=
∂∂=
∂∂=
∂∂=
=
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Formation of PDE by eliminating arbitrary constant
Let us consider the functional relationf(x, y, z, a, b) = 0 -------- (1)
Where a and b are arbitrary constant to be eliminated
Differentiating (1) partially with respect to x and y, we get
)3(00
)2(00
−−−−−−−=∂∂+
∂∂
⇒=∂∂
∂∂+
∂∂
−−−−−−−=∂∂+
∂∂
⇒=∂∂
∂∂+
∂∂
qz
f
y
f
y
z
z
f
y
f
pz
f
x
f
x
z
z
f
x
f
Equation (2) and (3) will contain a and b. If we eliminate a andb from (1), (2) and (3) we get the PDE (involving p and q) of the first order.
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Remarks:
�If the number of constants to be eliminated is equal to numberof independent variables, the PDE got after elimination will be of first order.
�If the number of constants to be eliminated is more than the number independent variables, the resulting PDE will be of second or higher order.
�Answer is not unique.
Problem 1Form the partial differential equation by eliminating a and b from
Solution:
Differentiating (1) partially w.r. t x and y we get
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))(( 2222 byaxz ++=
)1())(( 2222 −−−−−−++= byaxzGiven
))(2( 22 byxx
zp +=
∂∂=
)2(2
22 −−−−−−−+=⇒ byx
p
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Differentiating (1) partially w.r. t ‘y’ we get
)2)(( 22 yaxy
zq +=
∂∂=
)3(2
22 −−−−−−−+=⇒ axy
q
Substitute (2) and (3) in equation (1), we have
x
p
y
qz
2.
2=
pqzxyei =4.).(
7
Problem 2 Form the partial differential equation by eliminating the arbitrary constants a and b from
Solution:
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α2222 cot)()( zbyax =−+−
)1(cot)()( 2222 −−−−−=−+− αzbyaxGiven
Diff. eqn. (1) p.w.r.t. x, we get
α2cot20)(2x
zzax
∂∂=+−
)2(cot 2 −−−−−−−=−⇒ αpzax
Diff. eqn. (1) p.w.r.t. y, we get
α2cot2)(20y
zzby
∂∂=−+
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)3(cot 2 −−−−−−−=−⇒ αqzby
Substitute (2) and (3) in equation (1), we have
ααα 222222 cot)cot()cot( zqzpz =+
αα 222242 cot)(cot zqpz =+
1)(cot 222 =+ qpα
α222 tan.).( =+ qpei
9
Problem 3Form the partial differential equation by eliminating the arbitrary constants a and b from
Solution:
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bayxaz ++= 22
)1(22 −−−−−−−−++= bayxazGiven
Diff. eqn. (1) p.w.r.t. x, we get
)2(2 −−−−−−−=∂∂= a
x
zp
Diff. eqn. (1) p.w.r.t. y, we get
)3(2
2 −−−−−−−=⇒=∂∂=
y
qaay
y
zq
10
Problem 4Form the partial differential equation by eliminating the arbitrary constants a and b from
Solution:
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Substitute (3) in equation (2), we have
2
2
=
y
qp
224 qpy =
nn byaxz +=
)1(−−−−−−+= nn byaxzGiven
Diff. eqn. (1) p.w.r.t. x, we get
1−=∂∂= nxna
x
zp
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x
xnap
n
=
)2(−−−−−−−= nxan
xp
Diff. eqn. (1) p.w.r.t. y, we get
1−=∂∂= nynb
y
zq
y
ynbq
n
=
)3(−−−−−−−= nybn
yq
12
Problem 5Find the partial differential equation of all planes cutting equal intercepts from the x and y axes.
Solution:
The equation of the plane cutting equal intercept from x and y axes is
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Substitute (2) and (3) in equation (1), we have
n
yq
n
xpz +=
yqxpznei +=.).(
)1(1 −−−−−−=++c
z
a
y
a
x
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Diff. eqn. (1) p.w.r.t. x, we get
Diff. eqn. (1) p.w.r.t. y, we get
001 =++
c
p
a
)2(1 −−−−−−−−=ac
p
01
0 =++c
q
a
)3(1 −−−−−−−−=ac
q
Divide (2) by (3), we get
1=q
pqpei =.).(
14
Problem 6Find the partial differential equation of all planes passing through the origin
Solution:
The equation of the plane passing through the origin is
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ax + by + cz = 0
ybxazc −−=⇒
yc
bx
c
az −−=⇒
)1(.).( −−−−−−−+= yBxAzei
where A and B are arbitrary constants
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Problem 7Find the PDE of all planes which are at a constant distance ‘k’from the origin.
Diff. eqn. (1) p.w.r.t. x, we get
Diff. eqn. (1) p.w.r.t. y, we get
Ax
zp =
∂∂=
By
zq =
∂∂=
Substitute (2) and (3) in equation (1), we have
yqxpz +=
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Solution:
The equation of the plane having constant distance ‘k’ from the origin is
)1(0222 −−−−−−=++−++ cbakzcybxa
Diff. eqn. (1) p.w.r.t. x, we get
0=+ pca
)2(−−−−−−−−=⇒ pca
0=+ qcb
)3(−−−−−−−−=⇒ qcb
Substitute (2) and (3) in equation (1), we have
Diff. eqn. (1) p.w.r.t. y, we get
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Problem 8 Form the partial differential equation of all spheres whose centre lies on the z-axis.
Solution:
Any point on the z-axis is of the form (0, 0, a)
Then the equation of the sphere with centre (0, 0, a) and radius k (say) is
where ‘a’ is the arbitrary constant.
022222 =++−+−− cqcpckzcyqcxpc
0122 =++−+−− qpkzyqxp
1.).( 22 ++++= qpkyqxpzei
)1()( 2222 −−−−−−−=−++ kazyx
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Diff. eqn. (1) p.w.r.t. x, we get
Diff. eqn. (1) p.w.r.t. y, we get
0)(202 =−++ pazx
)2()( −−−−−−−−−= pazx
0)(220 =−++ qazy
)3()( −−−−−−−−−= qazy
Divide (2) by (3), we get
q
p
y
x =
..).( xqypei =
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Problem 9Find the partial differential equation of the family of spheres having their centres on the line x = y = z.
Since the centre (a, b, c) lies on the line x = y = z, we have a = b = c
Hence the equation of the sphere is
Solution:
(x – a)2 + (y – a)2 + (z – a)2 = r2 ---------------- (1)
where ‘a’ is the arbitrary constants.
Diff. eqn. (1) p.w.r.t. x, we get
0)(2)(2 =−+− pazax
)2()1(222 −−−−−−−+=+ papzx
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Diff. eqn. (1) p.w.r.t. y, we get
0)(2)(2 =−+− qazay
)3()1(222 −−−−−−−+=+ qaqzy
Divide (2) by (3), we get
q
p
qzy
pzx
++=
++
1
1
)(2
)(2
)1)(()1)(( pqzyqpzx ++=++
qpzqzpyyqpzpzqxx +++=+++
yxqxzpzyei −=−+− )()(.).(
21
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Formation of PDE by eliminating arbitrary functions
Let us consider the relation f (u, v) =0 --------(1)where u and v are functions of x, y ,z and f is an arbitrary function to be eliminated.
)2(0 −−−−−−−=
∂∂+
∂∂
∂∂+
∂∂+
∂∂
∂∂
pz
v
x
v
v
fp
z
u
x
u
u
f
),
,,,(
yxoffunctiona
turniniszandzyxoffunctionsarevanduSince
)3(0 −−−−−−−=
∂∂+
∂∂
∂∂+
∂∂+
∂∂
∂∂
qz
v
y
v
v
fq
z
u
y
u
u
f
Differentiating (1) partially with respect to x and y we get
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)3()2(lim,lim andfromv
fand
u
finateeusletfinatingeofInstead
∂∂
∂∂
zyxoffunctionsareRQPwhere
RQqPp
formtheofequationangetwe
,,,,
)4(−−−−−−=+
Remarks:
� Equation (4) is called Lagrange’s linear PDE whose solution will be discussed later.
� The order of PDE formed depends only on the number of arbitrary functions eliminated
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Problem 1Form the partial differential equation by eliminating an arbitrary function from
Solution:
)( 22 yxfz +=
)1()( 22 −−−−−−+= yxfzGiven
Diff. eqn. (1) p.w.r.t. x, we get
Diff. eqn. (1) p.w.r.t. y, we get
)2()2()( 22 −−−−−−+′= xyxfp
)3()2()( 22 −−−−−−+′= yyxfq
Divide (2) by (3), we get
y
x
q
p = xqypei =.).(
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Problem 2Form the partial differential equation by eliminating the arbitrary functions from
Solution:
).()( 21 yfxfz =
)1()()( 21 −−−−−−−= yfxfzGiven
Diff. eqn. (1) p.w.r.t. x, we get
)2()()( 21 −−−−−−′= yfxfp
Diff. eqn. (1) p.w.r.t. y, we get
)3()()( 21 −−−−−−′= yfxfqDiff. eqn. (2) p.w.r.t. x, we get
)4()()( 21 −−−−−−′′= yfxfr
Diff. eqn. (2) p.w.r.t. y, we get
)5()()( 21 −−−−−−′′= yfxfs
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Problem 3Form the partial differential equation by eliminating an arbitrary function from
Solution:
Diff. eqn. (3) p.w.r.t. y, we get
)6()()( 21 −−−−−−′′= yfxft
From (2) and (3) we have
)()()()( 2121 yfxfyfxfqp ′′=
szqpei =.).(
)( 22 yxfxyz ++=
)1()( 22 −−−−−−++= yxfxyzGiven
Diff. eqn. (1) p.w.r.t. x, we get
)2()( 22 xyxfyp +′+=26
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)2()2()( 22 −−−−−−+′=− xyxfyp
Diff. eqn. (1) p.w.r.t. y, we get
)2()( 22 yyxfxq +′+=
)3()2()( 22 −−−−−−+′=− yyxfxq
Divide (2) by (3), we get
y
x
xq
yp =−−
22 xxqyyp −=−22.).( xyxqypei −=−
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Problem 4Eliminate the arbitrary function ‘f ’ from the relation
Solution:
++= yx
fyz log1
22
)1(log1
22 −−−−−
++= yx
fyzGiven
)2(1
log1
202
−−−−−−−
−
+′+=x
yx
fp
+′+=y
yx
fyq1
log1
22
Diff. eqn. (1) p.w.r.t. x, we get
Diff. eqn. (1) p.w.r.t. y, we get
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)3(1
log1
22 −−−−−−−
+′=−y
yx
fyq
Dividing (2) by (3), we have
+′
−
+′=
−y
yx
f
xy
xf
yq
p
1log
12
1log
12
2
2
y
x
yq
p
/1
/1
2
2−=−
=>
22 x
y
yq
p −=−
=>
)2(2 yqypx −−==>22 2.).( yqypxei =+
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Problem 5Form the partial differential equation by eliminating the
arbitrary function from
Solution:
0,2 =
−z
xxyzφ
The given equation can be written as
)1(2 −−−−−−−
=−z
xfxyz
Diff. eqn. (1) p.w.r.t. x, we get
)2(.1.
22
−−−−−−
−
′=−z
pxz
z
xfypz
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Diff. eqn. (1) p.w.r.t. y, we get
)3(22
−−−−−−
−
′=−z
qx
z
xfxqz
Divide (2) by (3), we get
xq
xpz
xqz
ypz
−−=
−−
2
2
))(2())(2( xpzxqzxqypz −−=−−
22 222 xpxzxqpzqzqyxxqpz +−−=+−
xzqyxzpxei =−+ )2(.).( 22
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Problem 6Eliminate the arbitrary function ‘f ’ from the relation
Solution:
0),( 222 =++++ zyxzyxf
The given equation can be written as
)1()(222 −−−−−−−++=++ zyxzyx φDiff. eqn. (1) p.w.r.t. x, we get
)01()(202 pzyxpzx ++++′=++ φ
)2()1()(22 −−−−−−−+++′=+ pzyxpzx φDiff. eqn. (1) p.w.r.t. y, we get
)10()(220 qzyxqzy ++++′=++ φ
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)3()1()(22 −−−−−−−+++′=+ qzyxqzy φ
Dividing (2) by (3), we have
)1()(
)1()(
22
22
qzyx
pzyx
qzy
pzx
+++′+++′
=++
φφ
)1(
)1(
q
p
qzy
pzx
++=
++
)1)(()1)(( pzqyqzpx ++=++
qpzqzpyyqpzpzqxx +++=+++
yxqxzpzyei −=−+− )()(.).(
33
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Lagrange’s linear PDE:(Linear first order PDE)
The linear PDE of first order is known as Lagrange’s linear equation is of the form
Pp + Qq = R where P,Q, R are functions of x, y, z
This is got by eliminating arbitrary function f (u, v)=0 or u=F(v)
To solve Pp + Qq = R
1. Form the auxiliary equation of the form
R
dz
Q
dy
P
dx ==
2. Solve these auxiliary simultaneous equation, giving two independent solution u=C1 and v= C2
3. The general solution is f (u, v)=0 or u=F(v)
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Problem 1Find the solution of
Solution:
This is Lagrange’s linear PDE of the form Pp + Qq =R
R
zd
Q
yd
P
xdareEA ==..
222 z
zd
y
yd
x
xd ==
222 zqypx =+
Take 1st and 2nd ratio, we have
22 y
yd
x
xd =
Integrating, we get 1
11c
yx+−=−
1
11c
xy=−=>
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Take 2nd and 3rd ratio, we have
22 z
zd
y
yd =
Integrating, we get
2
11c
zy+−=−
2
11c
yz=−=>
Hence the required solution is
011
,11 =
−−
yzxyF
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Problem 2
Using multiplier 1/x, 1/y, 1/z and then add, each ratio is
Solution:
Solve: )()()( yxzqxzypzyx −=−+−
This is Lagrange’s linear PDE of the form Pp + Qq =R
R
zd
Q
yd
P
xdareEA ==..
)()()( yxz
zd
xzy
yd
zyx
xd
−=
−=
−
yxxzzy
z
zd
y
yd
x
xd
−+−+−
++0=++⇒
z
zd
y
yd
x
xd
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Using multiplier 1, 1, 1 and then add, each ratio is
getwegIntegratin
1loglogloglog czyx =++
1log)log( czyx =⇒
1czyx =⇒
zyzxyxyzxzxy
dzdydx
−+−+−++=
0=++⇒ dzdydx
getwegIntegratin 2czyx =++
Hence the required solution is 0),( =++ zyxzxyF
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Problem 3
Solution:
Solve: 22)( xyyqpxz −=−
This is Lagrange’s linear PDE of the form Pp + Qq =R
R
zd
Q
yd
P
xdareEA ==..
22 xy
zd
zy
yd
zx
xd
−=
−=
Take 1st and 2nd ratio, we have
zy
yd
zx
xd
−=
y
yd
x
xd
−=
Integrating, we get
1logloglog cyx +−=
1logloglog cyx =+
1.).( cyxei =39
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Using multiplier x,y,z and then add, each ratio is
zxzyzyzx
dzzdyydxx2222 −+−
++=
0=++⇒ dzzdyydxx
Integrating, we get
2
222
222c
zyx =++
2222.).( czyxei =++
Hence the required solution is
0),( 222 =++ zyxyxF
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Problem 4
Solution:
Using multiplier 1/x, 1/y, 1/z and then add, each ratio is
Solve: )()()( 222222 yxzqxzypzyx −=−+−
)()()( 222222 yxz
zd
xzy
yd
zyx
xd
−=
−=
−
222222 yxxzzy
z
zd
y
yd
x
xd
−+−+−
++=
This is Lagrange’s linear PDE of the form Pp + Qq =R
R
zd
Q
yd
P
xdareEA ==..
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Using multiplier x,y,z and then add, each ratio is
0=++⇒z
zd
y
yd
x
xd
getwegIntegratin
1loglogloglog czyx =++
1log)log( czyx =⇒
1czyx =⇒
222222222222 yzxzxyzyzxyx
dzzdyydxx
−+−+−++=
0=++⇒ dzzdyydxx
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Problem 5
Solution:
Integrating, we get
2
222
222c
zyx =++
2222.).( czyxei =++
Hence the required solution is
0),( 222 =++ zyxxyzF
Solve: mxlyqlznxpnymz −=−+− )()(
This is Lagrange’s linear PDE of the form Pp + Qq =R
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Using multiplier l,m,n and then add, each ratio is
R
zd
Q
yd
P
xdareEA ==..
mxly
zd
lznx
yd
nymz
xd
−=
−=
−
nmxnlylmzmnxnlylmz
ndzmdyldx
−+−+−++=
0=++⇒ ndzmdyldx
getwegIntegratin
1cnzmylx =++
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Using multiplier x,y,z and then add, each ratio is
xzmlyzlyzxynxynxzm
dzzdyydxx
−+−+−++=
0=++⇒ dzzdyydxx
Integrating, we get
2
222
222c
zyx =++
2222.).( czyxei =++
Hence the required solution is
0),( 222 =++++ zyxnzmylxF
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Problem 6
Solution:
Using multiplier 1/x, (-1/y), 1/z and then add, each ratio is
Solve: )()()( 2222 yxzqzxypzyx −=+++
)()()( 2222 yxz
zd
zxy
yd
zyx
xd
−=
+=
+
)()()( 2222 yxzxzy
z
zd
y
yd
x
xd
−++−+
+−=
This is Lagrange’s linear PDE of the form Pp + Qq =R
R
zd
Q
yd
P
xdareEA ==..
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Using multiplier x,(-y),-1 and then add, each ratio is
0=+−⇒z
zd
y
yd
x
xd
getwegIntegratin
1loglogloglog czyx =+−
1loglog)log( cyzx =−⇒
1.).( cy
zxei =
zyzxzyyxzxyx
dzdyydxx22222222 +−−−+
−−=
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0=−−⇒ dzdyydxx
Integrating, we get
2
22
22cz
yx =−−
222 2.).( czyxei =−−
Hence the required solution is
02, 22 =
−− zyx
y
zxF
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Problem 7
Solution:
Solve: xzqxypzyx 22)( 222 =+−−
xz
zd
xy
yd
zyx
xd
22222==
−−
This is Lagrange’s linear PDE of the form Pp + Qq =R
R
zd
Q
yd
P
xdareEA ==..
Take 2nd and 3rd ratio, we have
xz
zd
yx
yd
22=
z
zd
y
yd ==>
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Using multiplier x,y,z and then add, each ratio is
Integrating, we get
1logloglog czy +=
1logloglog czy =−
1.).( cz
yei =
xzxyzyxx
dzzdyydxx22222 22)( ++−−
++=
xzxyx
dzzdyydxx223 ++
++=)( 222 zyxx
dzzdyydxx
++++=
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Equate this to 2nd ratio, we have
xy
dy
zyxx
dzzdyydxx
2)( 222=
++++
y
dy
zyx
dzzdyydxx
2222=
++++
⇒
getwegIntegratin
2222 loglog
2
1)log(
2
1cyzyx +=++
2222 loglog)log( cyzyx +=++
2222 loglog)log( cyzyx =−++
2
222
.).( cy
zyxei =++
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Problem 8
Solution:
Hence the required solution is
0,222
=
++y
zyx
z
yF
Solve: yxzqxzypzyx −=−+− 222 )()(
This is Lagrange’s linear PDE of the form Pp + Qq =R
R
zd
Q
yd
P
xdareEA ==..
yxz
zd
xzy
yd
zyx
xd
−=
−=
− 222
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(Subtracting 1st and 2nd ratio)
(Subtracting 2nd and 3rd ratio)
-----------(A)
)()( 22 xzyzyx
dydxratioEach
−−−−=
)()(
)(22 zyxzyx
yxd
−+−−=
)())((
)(
yxzyxyx
yxd
−++−−=
))((
)(
zyxyx
yxd
++−−=
)()( 22 yxzxzy
dzdyratioEach
−−−−=
)()(
)(22 xzyxzy
zyd
−+−−=
53
6/4/2015B.Saravanan
---------(B)
From (A) and (B) we have
)())((
)(
zyxzyzy
zyd
−++−−=
))((
)(
zyxzy
zyd
++−−=
))((
)(
))((
)(
zyxzy
zyd
zyxyx
yxd
++−−=
++−−
)(
)(
)(
)(
zy
zyd
yx
yxd
−−=
−−
⇒
Integrating we get
1log)log()log( czyyx +−=−1.).( c
zy
yxei =
−−
54
6/4/2015B.Saravanan
Using multiplier 1,1,1 and then add, each ratio is
Using multiplier x,y,z and then add, each ratio is
-------(C)
---------(D)
xzzyyxzyx
dzdydxratioEach
−−−++++= 222
xzzyyxzyx
zyxd
−−−++++=
222
)(
zyxzyx
dzzdyydxxratioeach
3333 −++++=
))(( 222 xzzyyxzyxzyx
dzzdyydxx
−−−++++++=
55
6/4/2015B.Saravanan
From (C) and (D)
xzzyyxzyx
zyxd
−−−++++
222
)(
))(( 222 xzzyyxzyxzyx
dzzdyydxx
−−−++++++=
dzzdyydxxzyxdzyx ++=++++ )()(
Integrating we get
2
2222
2222
)(c
zyxzyx +++=++
22222)( czyxzyx +++=++
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6/4/2015B.Saravanan
2222222 )(2 czyxxzzyyxzyx +++=+++++
2)(2 cxzzyyx =++
2.).( cxzzyyxei =++
Hence the required solution is
0, =
++
−−
xzzyyxzy
yxF
57
6/4/2015B.Saravanan 58
Non-Linear first order PDE (Standard types)
Those equations in which p and q occur other than the first degree and product of p, q terms are called non linear first order PDE
Ex:p2 +q2+pq = 4
Types of Solutions:
1. A solution in which the number of arbitrary constants is equal to number of independent variable is called complete integral or complete solution.
2. In the complete integral if we give particular value to arbitrary constant we get particular integral
6/4/2015B.Saravanan 59
)3(0
)2(0
0,)1(.
)1(0),,,,(
0),,,,(.3
−−−−−=∂∂
−−−−−=∂∂
−−−−−==
b
a
getwetoequateandbatorespectwithpartiallyDiff
bazyx
issolutioncompletewhosePDEabeqpzyxfLet
φ
φ
φ
Eliminate a and b from (1),(2) and (3), the resulting one is called singular integral
6/4/2015B.Saravanan 60
Standard types of first order PDE
f (p ,q)=0 ( i.e. equations containing p and q only)
Type I
Its complete integral is given by z = a x + b y + c -------- (1)
where a and b are connected by f (a, b)=0----------(2)
From (2) express b as function of a
tconsarbitraryarecandawhere
cyaaxz
egralcompletegetweeqninsubstituteandabei
tan
)3()()1(
int)1.()(..
−−−−−−−++=⇒
=φ
φ
6/4/2015B.Saravanan 61
absurdiswhichc
zand
a
z
getwetoequateand
catorespectwithpartiallyDiffegralgularfindTo
0100
0
,)3(.intsin
=⇒=∂∂=
∂∂
Hence there is no singular integral
)()()(0
)(.
)()()(
int)(
Bagyax
getwe
atorespectwithpartiallyADiffand
Aagyaaxz
getweegralcompleteinagcputsolutiongeneralfindTo
−−−−−−−′+′+=
−−−−−++==
φ
φ
Eliminate ‘a’ from (A) and (B) we get general integral or general solution.
6/4/2015B.Saravanan
Problem 1
Find the complete integral of 1=+ qp
Solution:
)1(1 −−−−−=+ qpGiven
The complete solution of equation (1) is
cybxaz ++=
where 1=+ ba ab −=⇒ 1 ( )21 ab −=⇒
Hence the complete integral is
( ) cyaxaz +−+=2
1
This is of the form f(p,q)=0
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6/4/2015B.Saravanan
Problem 2
Find the complete integral of p-q=0
Solution:
The complete solution of equation (1) is
Given p – q = 0 ------------ (1)
cybxaz ++=
Where a-b=0 => b=a
Hence the complete integral is
cyaxaz ++=
This is of the form f(p,q)=0
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6/4/2015B.Saravanan
Problem 3
Find the complete integral of 0422 =−+ pqqp
Solution:
)1(0422 −−−−−−−=−+ pqqpGiven
The complete solution of equation (1) is cybxaz ++=
where 0422 =−+ baba
04 22 =+−⇒ abab
1.2
.1.4164 22 aaab
−±=
This is of the form f(p,q)=0
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6/4/2015B.Saravanan
2
124 2aa ±=
2
324 aa ±=
)32( ±= a
cyaxaz +±+= )32(
Hence the complete integral is
65
6/4/2015B.Saravanan 66
f (p, q, z)=0 ( i.e. equations containing p ,q and z only)
Type II
The given PDE is f (p, q, z)=0--------(1)
Put q=ap in equation (1) and find p and q as function of z
Substitute p and q in dz = p dx + q dy
(keep z terms in LHS and remaining in RHS)
On integrating we get the complete integral
Procedure for obtaining Singular integral and general solution are same as explained in type I
6/4/2015B.Saravanan
Problem 1
Solve: qzqp =+ )1(
Solution:
This is of the form f(z , p, q) = 0
)1()1( −−−−=+ qzqpGiven
Let q = ap
Then equation (1) becomes
p(1 + ap) = ap z
1 + ap = aza
zap
1−=⇒
paqNow =,
−=a
zaa
1
1−= za
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6/4/2015B.Saravanan
Substitute p and q in the relation
dz = p dx + q dy
ydzaxda
zadz )1(
1 −+−=
yda
xd
za
zd +=−1
getwegIntegratin ,
bya
x
a
za ++=− )1log(
)2()1log(.).( −−−−−−++=− byaxzaei
which is the complete integral
68
6/4/2015B.Saravanan
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, in turn, we get
yza
a =−1
10 =and
The last equation is absurd and shows that there is no singular integral.
To find general integral, assume b = f(a)
Then equation (2) becomes
)3()()1log()2( −−−−−++=−=> afyaxza
Diff. eqn. (3) p.w.r.t. ‘a’, we get
)4()(1
−−−−−−−−′+=−
afyza
a
The eliminant of ‘a’ between equations (3) and (4) gives the general integral.
69
6/4/2015B.Saravanan 70
(i.e. equations containing p , q , x and y)
Type III
f1 (x, p) = f2 (y, q)
Let f1 (x, p) = f2 (y, q) = a
f1 (x, p) = a and f2 (y, q) = a
solve for p and q (write p as function of x and q as function of y)
p=f(x) and q=g(y)
Substitute p and q in dz = pdx + qdy
=> dz= f(x) dx +g(y )dy
6/4/2015B.Saravanan 71
On integrating we get
∫∫ ++= bdyygdxxfz )()(
Which is the complete integral contains two arbitrary constant a and b
Procedure for obtaining Singular integral and general solution are same as explained in type I
6/4/2015B.Saravanan
Problem 1
Solve: 2222 yxqp +=+
Solution:
)1(2222 −−−−−+=+ yxqpGiven
This is of the form f(x , p) = g( y , q)
22222 aqyxp =−=−⇒
22222 axpaxp +±=⇒=−
22222 ayqaqy −±=⇒=−
2222 qyxp −=−⇒
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6/4/2015B.Saravanan
Substitute p and q in the relation
dz = p dx + q dy
dyaydxaxdz 2222 −±+±=getwegIntegratin
)2(cosh22
sinh22
12
22
12
22
−−−−+
−−±
++±=
−
−
ba
yaay
y
a
xaax
xz
which is the complete integral
73
6/4/2015B.Saravanan
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, in turn, we get
10
).(cosh1)/(
1
22
)2(
2
).(sinh)/(1
1
22
2
20
122
2
22
122
2
22
=
−
+
−−
−−
−±
+
−+
++
±=
−
−
and
aa
y
a
y
ax
a
ay
ay
aa
x
a
x
ax
a
ax
ax
The last equation is absurd and shows that there is no singular integral
To find general integral, assume b = f(a)
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6/4/2015B.Saravanan
Then equation (2) becomes
)3()(cosh22
sinh22
12
22
12
22
−−−−−−+
−−±
++±=
−
−
afa
yaay
y
a
xaax
xz
Diff. eqn. (3) p.w.r.t. ‘a’, we get
)4()().(cosh1)/(
1
22
)2(
2
).(sinh)/(1
1
22
2
20
122
2
22
122
2
22
−−−′+
−
+
−−
−−
−±
+
−+
++
±=
−
−
afaa
y
a
y
ax
a
ay
ay
aa
x
a
x
ax
a
ax
ax
The eliminant of ‘a’ between equations (3) and (4) gives the general integral.
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6/4/2015B.Saravanan
Problem 2
Find the complete integral of xpq =
Solution:
This is of the form f(x , p) = g( y , q)
)1(−−−= xpqGiven
Let q = a
Then equation (1) becomes
xap =a
xp =⇒
Substitute p and q in the relation
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6/4/2015B.Saravanan
dyqdxpdz +=
dyadxa
xdz +=
getwegIntegratin ,
baya
xz ++=
2
2
which is the complete integral.
77
6/4/2015B.Saravanan 78
Type IV (Clairaut’s form)
An equation of the form z = p x + q y + f (p, q) is known as Clairaut’s equation
Its complete integral is z = a x + b y + f (a, b) ------(1) (by replacing p by a and q by b)
To find singular integral diff. (1) partially with respect to a, b we get
)3(0
)2(0
−−−−−−−∂∂+=
−−−−−−−∂∂+=
b
fy
a
fx
Eliminate a and b from (1), (2) and (3) we get singular integral. Procedure for obtaining general solution are same as explained in type I
6/4/2015B.Saravanan
Problem 1
Find the complete integral of pqp
y
q
x
pq
z ++=
Solution:
pqp
y
q
x
pq
zGiven ++=
)1(−−−−−−++= pqqpyqxpz
The complete integral of equation (1) is
babaybxaz ++= (replacing p by a and q by b)
This is in Clairaut’s form
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6/4/2015B.Saravanan
Problem 2
Find the singular integral of z=p x +q y +p q
Solution: Given z=p x +q y +p q
This is in Clairaut’s form
The complete integral of equation is
z= a x + b y + a b -------(1)
To find the singular integral, diff. (1) partially w.r.to a and b
0 = x + b => b = -x
0 = y + a => a = -y
(1)=> z= -x y – x y + x y => z= -x y
(replacing p by a and q by b)
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6/4/2015B.Saravanan
Problem 3
Find the singular solution of 22 qpqpqypxz ++++=
Solution:
)1(22 −−−−−++++= qpqpqypxzGiven
This is in Clairaut’s form
The complete integral of equation (1) is
)2(22 −−−−−−++++= bababyaxz(replacing p by a and q by b)
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, we get
bax ++= 20 )3(2 −−−−−−−=+⇒ xba
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6/4/2015B.Saravanan
bayand 20 ++= )4(2 −−−−−−−=+⇒ yba
Solving (3) and (4) we get
3
223
xyaxya
−=⇒−=3
223
yxbyxband
−=⇒−=
Substitute the values of a and b in equation (2) we have
2
2
3
2
3
2
3
2
3
2
3
2
3
2
−+
−
−+
−+
−+
−=
yxyxxy
xyyxy
xyxz
22)2( bababyaxz ++++==>
82
6/4/2015B.Saravanan
2
2
)2()2)(2(
)2()2(3)2(39
yxyxxy
xyyxyxyxz
−+−−+−+−+−=
22 3339 yxxyz −−=223.).( yxxyzei −−=
Problem 4
Find the singular integral of the partial differential equation
22 qpqypxz −++=
Solution:
)1(22 −−−−−−++= qpqypxzGiven
This is in Clairaut’s form
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6/4/2015B.Saravanan
The complete integral of equation (1) is
)2(22 −−−−−−−++= babyaxz(replacing p by a and q by b)
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, we get
ax 20 += )3(2
−−−−−−−=⇒x
a
by 20 −= )4(2
−−−−−−=⇒y
b
Substitute the values of a and b in equation (2) we have
22
2222)2(
−
−+
+
−==> yxyy
xxz
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6/4/2015B.Saravanan
2222 224 yxyxz −++−=224.).( xyzei −=
Problem 5
Find the singular integral of pqqypxz 2++=
Solution:
This is in Clairaut’s form
)1(2 −−−−−−++= pqqypxzGiven
The complete integral of equation (1) is
)2(2 −−−−−−−−++= baybxaz
(replacing p by a and q by b)
85
6/4/2015B.Saravanan
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, we get
)(2
20 b
abx += )3(−−−−−−=−⇒
a
bx
)(2
20 a
abyand += )4(−−−−−−=−⇒
b
ay
Multiplying (3) and (4) we get
x y=1, which is the singular integral.
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6/4/2015B.Saravanan
Problem 6
Solve: 221 qpqypxz ++++=
Solution:
)1(1 22 −−−−−++++= qpqypxzGiven
This is in Clairaut’s form
)2(1 22 −−−−−−−++++= babyaxz
The complete integral of equation (1) is
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, we get
(replacing p by a and q by b)
87
6/4/2015B.Saravanan
)2(12
10
22a
bax
+++=
)3(1 22
−−−−−−++
−=⇒ba
ax
)2(12
10
22b
bayand
+++=
)4(1 22
−−−−−−++
−=⇒ba
by
Substitute (3) and (4) in equation (2), we get
22
22
2
22
2
111
baba
b
ba
az +++
++−
++−=
221)2( babyaxz ++++==>
88
6/4/2015B.Saravanan
22
2222
1
1
ba
baba
+++++−−=
221
1
ba ++=
)5(1
1.).(
222 −−−−−−
++=
bazei
Squaring and adding (3) and (4), we have
22
2
22
222
11 ba
b
ba
ayx
+++
++=+
22
22
1
1)1(
ba
ba
++−++=
221
11
ba ++−=
89
6/4/2015B.Saravanan
which is the singular integral.
])5(sin[1 222 guzyx −=+
1.).( 222 =++ zyxei
To find general integral, assume b = f(a)
Then equation (2) becomes
)6()}({1)( 22 −−−−−−−−++++= afayafxaz
Diff. eqn. (6) p.w.r.t. ‘a’, we get
)7(]1).().(22[)}({12
1)(0
22−−−′+
+++′+= afafa
afayafx
The eliminant of ‘a’ between equations (6) and (7) gives the general integral.
90
6/4/2015B.Saravanan
Problem 7
Find the complete and singular solutions of
−++= p
p
qqypxz
Solution:
)1(−−−−−
−++= p
p
qqypxzGiven
This is in Clairaut’s form
The complete integral of equation (1) is
)2(−−−−−−−
−++= aa
bbyaxz
(replacing p by a and q by b)
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6/4/2015B.Saravanan
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, we get
102
−−=a
bx )3(1
2−−−−−−=−⇒
a
bx
ayand
10 += )4(
11 −−−−−−−=⇒−=⇒y
aa
y
Substitute (4) in (3) , we get
21
1
−
=−
y
bx
21 ybx =−⇒ )5(1
2−−−−−−−=⇒
y
xb
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6/4/2015B.Saravanan
Substitute (4) and (5) in equation (2), we have
+−−+
−+−=yy
x
y
xy
y
xz
1)1(12
y
xxxz
1)1(1 +−−−+−=
xzyei −= 1.).(
which is the singular integral
−++==> aa
bbyaxz)2(
93
6/4/2015B.Saravanan
Problem 1
Find the complete solution of 2zpqxy =
Solution:
)1()()( 2 −−−−= zqypxGiven
yYxXPut log,log ==
x
X
X
z
x
zp
∂∂
∂∂=
∂∂= .
xX
z 1
∂∂=
X
zpx
∂∂=⇒
X
zPwherePpxei
∂∂==.).(
94
6/4/2015B.Saravanan
y
Y
Y
z
y
zq
∂∂
∂∂=
∂∂= .
yY
z 1
∂∂=
Y
zqy
∂∂=⇒
Y
zQwhereQqyei
∂∂==.).(
Equation (1) becomes
)2(2 −−−−−= zQP
2)()()1( zqypx ==>
Let Q = aP Then equation (2) becomes
2.)2( zaPP ==>a
zP ==>
95
6/4/2015B.Saravanan
Substitute P and Q in the relation
dz = P dX + Q dY
YdzaXda
zdz +=
PaQNow =,
=a
za za=
YdaXdz
zda +=
getwegIntegratin ,
baYXza ++=log
byaxzaei ++= logloglog.).(
which is the complete solution.
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6/4/2015B.Saravanan
Problem 2
yxqpz +=+ )( 222
Solution:
Find the general solution of
)1()()( 22 −−−−−+=+ yxqzpzGiven
211 zzZPut == +
x
zz
x
Z
∂∂=
∂∂
2x
ZPwherepz
P
∂∂==⇒
2
y
zz
y
Z
∂∂=
∂∂
2y
ZQwhereqz
Q
∂∂==⇒
2
Equation (1) becomes yxQP +=
+
22
22
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6/4/2015B.Saravanan
)(4.).( 22 yxQPei +=+
aQyxP =−=−⇒ 22 44
axPaxPLet +±=⇒=− 442
ayQaQyAlso −±=⇒=− 44 2
Substitute p and q in the relation
dz = p dx + q dy
dyaydxaxdz −±+±= 44
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6/4/2015B.Saravanan
getwegIntegratin
bayax
z +−±+±=)2/3(4
)4(
)2/3(4
)4( 2/32/3
which is the complete integral.
To find singular integral, Diff. eqn. (2) p.w.r.t. ‘a’ and ‘b’, in turn, we get
)2(6
)4(
6
)4( 2/32/3
−−−−−−+−±+±= bayax
z
10)4(4
1)4(
4
10 2/12/1 =−±+±= andayax
The last equation is absurd and shows that there is no singular integral.
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6/4/2015B.Saravanan
To find general integral, assume b = f(a)
Then equation (2) becomes
)3()(6
)4(
6
)4( 2/32/3
−−−−−+−±+±= afayax
z
Diff. eqn. (3) p.w.r.t. ‘a’, we get
)4()()4(4
1)4(
4
10 2/12/1 −−−−−′+−±+±= afayax
The eliminant of ‘a’ between equations (3) and (4) gives the general integral.
100
6/4/2015B.Saravanan 101
Linear PDE of second and higher order with constant coefficients:
� Homogeneous PDE� Non-Homogeneous PDE
Homogeneous linear PDE
An equation in which the partial derivatives occurring are all of same order and coefficients are constant is called homogeneous PDE
Ex:
08423
3
2
3
2
3
3
3
=∂∂+
∂∂∂−
∂∂∂−
∂∂
y
z
yx
z
yx
z
x
z
0)2( 23 =′− zDDD
6/4/2015B.Saravanan 102
Standard form:
The standard form of homogeneous PDE of nth order with constant coefficients is of the form
DandDinreenthofpolinomialogeneousisDDfwhere
yxFzDDf
yDand
xDwhere
yxFzDaDDaDDaDa nn
nnn
′′−−−−−=′∂∂=′
∂∂=
=′++′+′+ −−
deghom),(
)1(),(),(
),()............( 222
110
The general solution of equation (1) is z=C.F+P.I
6/4/2015B.Saravanan 103
Procedure to find C.F:
C.F is the solution of equation
0............
1
0............
22
110
222
110
=++++⇒
=′==′++′+′+
−−
−−
nnnn
nn
nnn
amamama
DandmDput
DaDDaDDaDa
0),( =′DDf
The above equation is called auxiliary equation and solving the A.E we get n roots m1, m2, m3, ......... mn,
Case I (All the roots are distinct)
)(...........)()(. 2211 xmyfxmyfxmyfFC nn ++++++=
6/4/2015B.Saravanan 104
Case II (All the roots are equal)
)(..).........()()(. 11
132
1211 xmyfxxmyfxxmyxfxmyfFC nn +++++++= −
Note: roots may be real or complex
Particular Integral:
(1) R.H.S is an exponential function then
0),(),(
1
),(
1. ≠=
′= ++ bafife
bafe
DDfIP byaxbyax
....
.0),(
proceduresametheapplyandDtorwDrthe
atedifferentiandNrtheinxmultiplythenbafIf =
6/4/2015B.Saravanan 105
(2) R.H.S is an Trigonometric function then
)()(),,(
1.
22byaxCosorSin
DDDDfIP +
′′=
0),,()()(),,(
1 2222 ≠−−−+
−−−= babafifbyaxCosorSin
babaf
....
.0),,( 22
proceduresametheapplyandDtorwDrthe
atedifferentiandNrtheinxmultiplythenbabafif =−−−
(3) R.H.S is an Polynomial function then
nmnm yxDDyxDDf
IP 1)],(1[),(
1. −′+=
′= φ
6/4/2015B.Saravanan 106
(4) Exponential shift rule
),(),(
1),(
),(
1. yx
bDaDfeyxe
DDfIP byaxbyax φφ
+′+=
′= ++
(5) General rule
),(),(
1.
),(
yxFDDf
IP
functionotheranyisyxFIf
′=
byfactoreachoperatethenandDDffactorize ),( ′
dxmxaxFyxFDmD ∫ −=
′−),(),(
1
mxybyreplacedisaegrationafter +int
6/4/2015B.Saravanan
Problem 1
Solve 0)2( 23 =′− zDDD
Solution:
A.E. is m3 – 2m2 = 0 [Put D = m and D′ = 1]
m2(m – 2) = 0
m2 = 0 (or) m – 2 = 0
m = 0, 0, 2
)2()()( 321 xyfyfxyfz +++=∴
107
6/4/2015B.Saravanan
Problem 2
Solve 0)( 3 =′− zDD
Solution:
0)( 3 =′− zDD
A.E. is (m – 1)3 = 0 [Put D = m and D′ = 1]
(m – 1)(m – 1)(m – 1) = 0
m = 1, 1, 1
)()()( 32
21 xyfxxyfxxyfz +++++=∴
108
6/4/2015B.Saravanan
Problem 3
0)2( 22 =′+′− zDDDDSolve
Solution:
A.E. is m2 – 2m + 1 = 0 [Put D = m and D′ = 1]
(m – 1)(m – 1) = 0
m = 1, 1
)()( 21 xyfxxyfz +++=∴
109
6/4/2015B.Saravanan
Problem 4
0)( 3223 =′−′−′+ zDDDDDDSolve
Solution:
A.E. is m3 + m2 – m – 1 = 0 [Put D = m and D′ = 1]
m2(m + 1) –1(m + 1) = 0
(m + 1)(m2 – 1) = 0
m = –1, m2 = 1
1±=m
∴
m = 1, –1, –1
)()()( 321 xyfxxyfxyfz −+−++=∴
110
6/4/2015B.Saravanan
Problem 5
Solve 08423
3
2
3
2
3
3
3
=∂∂+
∂∂∂−
∂∂∂−
∂∂
y
z
yx
z
yx
z
x
z
Solution: The given equation can be written as
0)842( 3223 =′+′−′− zDDDDDD
A.E. is m3 – 2m2 – 4m + 8 = 0 [Put D = m and D′ = 1]
m2(m – 2) – 4(m – 2) = 0
(m – 2)(m2 – 4) = 0
m = 2, m2 = 4
2±=m
∴
=> m = 2, 2, –2
)2()2()2( 321 xyfxyfxxyfz −++++=∴
111
6/4/2015B.Saravanan
Problem 6
Solve )2sin()67( 2323 yxezDDDD yx ++=′−′− +
Solution:
A.E. is m3 – 7m – 6 = 0 [Put D = m and D′ = 1]
m = –1 is a root
The other roots are
m2 – m – 6 = 0
(m – 3)(m + 2) = 0
m = 3, –2
m = –1, –2, 3 )3()2()(. 321 xyfxyfxyfFC ++−+−=
112
6/4/2015B.Saravanan
yxeDDDD
IP +
′−′−= 2
3231 67
1.
yxe +
−−= 2
323 )1(6)1)(2(7)2(
1
yxe +−= 2
12
1
)2sin(67
1.
3232 yxDDDD
IP +′−′−
=
)2sin()4(6)4(7
1yx
DDD+
′−−−−−=
)2sin(2427
1yx
DD+
′+=
113
6/4/2015B.Saravanan
)2sin()89(3
1yx
DD+
′+=
)2sin()89)(89(3
89yx
DDDD
DD +′−′+
′−=
)2sin()6481(3
8922
yxDD
DD +′−
′−=
)2sin()]4(64)1(81[3
89yx
DD +−−−
′−=
525
)]2[sin(8)]2[sin(9 yxDyxD +′−+=
114
6/4/2015B.Saravanan
)]2cos(16)2cos(9[525
1yxyx +−+=
)]2cos(7[525
1yx +−=
)2cos(75
1yx +−=
∴
z = C.F + P.I1 + P.I2
)2cos(75
1
12
1
)3()2()(.).(
2
321
yxe
xyfxyfxyfzei
yx +−−
++−+−=
+
115
6/4/2015B.Saravanan
Problem 7
)cos()( 23223 yxezDDDDDD yx ++=′−′−′+ +Solve
Solution:
A.E. is m3 + m2 – m – 1 = 0 [Put D = m and D′ = 1]
m2(m + 1) –1(m + 1) = 0
(m + 1)(m2 – 1) = 0
m = –1, m2 = 1
1±=m
m = 1, –1, –1
)()()(. 321 xyfxxyfxyfFC −+−++=116
6/4/2015B.Saravanan
yxeDDDDDD
IP +
′−′−′+= 2
32231
1.
yxe +
−−+= 2
3223 )1()1)(2()1()2()2(
1
yxe += 2
9
1
)cos(1
.32232 yx
DDDDDDIP +
′−′−′+=
)cos(1
yxDDDD
+′++′−−
=
Since the denominator = 0, we have to multiply x on Nr. and Diff. Dr. w.r.t.‘D’
)cos(0
1yx +=
117
6/4/2015B.Saravanan
)cos(23 22
yxDDDD
x +′−′+
=
)cos()1()1(2)1(3
yxx +
−−−+−=
)cos(4
yxx +−=
z = C.F + P.I1 + P.I2
)cos(49
1
)()()(.).(
2
321
yxx
e
xyfxxyfxyfzei
yx +−+
−+−++=
+
118
6/4/2015B.Saravanan
Problem 8
Solve xezDDDD yx sin)44( 2322 +=′+′− −
Solution:
A.E. is 4m2 – 4m + 1 = 0 [Put D = m and D′ = 1]
(2m – 1)2 = 0
2
1,
2
1=m
++
+= xyfxxyfFC2
1
2
1.. 21
P.I1 = yxe
DDDD23
22 44
1 −
′+′−119
6/4/2015B.Saravanan
yxe 2322 )2()2)(3(4)3(4
1 −
−+−−=
yxe 23
64
1 −=
P.I2 = )0sin(44
122
yxDDDD
+′+′−
)0sin(00)1(4
1yx +
+−−=
xsin4
1−=
z = C.F + P.I1 + P.I2
xexyfxxyfzei yx sin4
1
64
1
2
1
2
1.).( 23
21 −+
++
+= −
120
6/4/2015B.Saravanan
Problem 9
yxeyxzDDDD −+=′+′+ 222 )2(Solve
Solution:
A.E. is m2 + 2m + 1 = 0 [Put D = m and D′ = 1]
(m + 1)(m + 1) = 0
m = –1, –1
)()(.. 21 xyfxxyfFC −+−=
P.I1 = yxe
DDDD−
′+′+ 22 2
1
yxe −
−+−+=
22 )1()1)(1(2)1(
1 Since the denominator = 0, we have to multiply x on Nr. and Diff. Dr. w.r.t.‘D’
121
6/4/2015B.Saravanan
yxeDD
x −
′+=
22
yxex −=2
2
P.I2 = yxDDDD
222 2
1′+′+
yx
D
DDDD
2
2
22 2
1
1
′+′+
=
yxD
DDD
D2
1
2
2
2
21
1−
′+′+= yx
D
DDD
D2
2
2
2
21
1
′+′−=
122
6/4/2015B.Saravanan
yxD
D
D2
2
21
1
′−=
′−= )(
2)(
1 222
yxD
Dyx
D
−= )(21 22
2x
Dyx
D
−=
3
21 32
2
xyx
D
−=
12
2
3
1 43 xyx
D
123
6/4/2015B.Saravanan
3012
54 xyx −=
z = C.F + P.I1 + P.I2
30122)()(.).(
542
21
xyxe
xxyfxxyfzei yx −++−+−= −
Problem 10
Solve yxzDDDD sin)43( 22 +=′−′+
Solution:
A.E. is m2 + 3m – 4 = 0 [Put D = m and D′ = 1]
(m – 1)(m + 4) = 0
m = 1, – 4 )4()(.. 21 xyfxyfFC −++==>124
6/4/2015B.Saravanan
P.I1 = xDDDD 22 43
1′−′+
x
D
DDDD
′−′+
=
2
22 43
1
1
xD
DDD
D
1
2
2
2
431
1−
′−′+=
xD
DDD
D
′−′−= 2
2
2
431
1
[ ]01
2−= x
D
=
2
1 2x
D 6
3x=
125
6/4/2015B.Saravanan
P.I2 = )0sin(43
122
yxDDDD
+′−′+
)0sin()1(400
1yx +
−−+=
ysin4
1=
z = C.F + P.I1 + P.I2
yx
xyfxyfzei sin4
1
6)4()(.).(
3
21 ++−++=
126
6/4/2015B.Saravanan
Problem 11
Solve xyyxy
z
yx
z
x
z ++=∂∂−
∂∂∂+
∂∂
)sinh(22
22
2
2
Solution:
The given equation can be written as
xyyxzDDDD ++=′−′+ )sinh()2( 22
A.E. is m2 + m – 2 = 0 [Put D = m and D′ = 1]
(m + 2)(m – 1) = 0
m = –2, 1
)()2(.. 21 xyfxyfFC ++−=
127
6/4/2015B.Saravanan
P.I1 = )sinh(2
122
yxDDDD
+′−′+
−′−′+
=+−+
22
1 )(
22
yxyx ee
DDDD
′−′+
−′−′+
= −−+ yxyx eDDDD
eDDDD 2222 2
1
2
1
2
1
−−−−+−−
−+= −−+ yxyx ee
2222 )1(2)1)(1()1(
1
)1(2)1)(1()1(
1
2
1
Since the denominator = 0, we have to multiply x on Nr. and Diff. Dr. w.r.t. ‘D’ we get
128
6/4/2015B.Saravanan
yxyx ex
ex −−+ +=
66
−−−
+= −−+ yxyx e
xe
x
12122
1
′+
−′+
= −−+ yxyx eDD
xe
DD
x
222
1
P.I2 = xyDDDD 22 2
1′−′+
xy
D
DDDD
′−′+
=
2
22 2
1
1
129
6/4/2015B.Saravanan
xyD
DDD
D
1
2
2
2
21
1−
′−′+=
xyD
DDD
D
′−′−=
2
2
2
21
1
xyD
D
D
′−= 1
12
′−= )()(
12
xyD
Dxy
D
−= )(11
2x
Dxy
D
130
6/4/2015B.Saravanan
z = C.F + P.I1 + P.I2
24666)()2(.).(
43
21
xyxe
xe
xxyfxyfzei yxyx −+++++−= −−+
246
43 xyx −=
−=
62
1 32 xyx
D
−=
2
1 2
2
xxy
D
131
6/4/2015B.Saravanan
Problem 12
Solve xyy
z
yx
z
x
zsin65
2
22
2
2
=∂∂+
∂∂∂−
∂∂
Solution:
The given equation can be written as
xyzDDDD sin)65( 22 =′+′−
A.E. is m2 – 5m + 6 = 0 [Put D = m and D′ = 1]
(m – 2)(m – 3) = 0
m = 2, 3
)3()2(.. 21 xyfxyfFC +++=
132
6/4/2015B.Saravanan
P.I = xyDDDD
sin65
122 ′+′−
xyDDDD
sin)3()2(
1′−′−
=
′−′−
= xyDDDD
sin3
1
2
1
∫ −′−
= dxxxcDD
sin)3(2
1
[ ])sin)(3()cos)(3(2
1xxxc
DD−−−−−
′−=
where y = c – 3x
133
6/4/2015B.Saravanan
z = C.F + P.I
xyxxyfxyfzei sincos5)3()2(.).( 21 −++++=
xyx sincos5 −=
xxxy cos3cos2sin ++−=
)cos(3)]cos)(2())(sin2([ xxxxc −−−−−−−=
∫ −−−= dxxxxc ]sin3cos)2([
]sin3cos[2
1xxy
DD−−
′−=
where y = c – 2x
134
6/4/2015B.Saravanan
Problem 13
Solve )32sin()( 22 yxezDD yx +=′− −
Solution:
A.E. is m2 – 1 = 0 [Put D = m and D′ = 1]
m2 = 1
1±=m
)()(.. 21 xyfxyfFC −++=
P.I = )32sin(1
22yxe
DDyx +
′−−
)32sin()1()1(
122 yx
DDe yx +
−′−+= −
135
6/4/2015B.Saravanan
)32sin(1212
122
yxDDDD
e yx +−′+′−++
= −
)32sin(22
122
yxDDDD
e yx +′+′−+
= −
)32sin(2)9(24
1yx
DDe yx +
′+−−+−= −
)32sin(5)(2
1yx
DDe yx +
+′+= −
)32sin(]5)(2][5)(2[
]5)(2[yx
DDDD
DDe yx +
−′++′+−′+= −
136
6/4/2015B.Saravanan
)32sin(25)(4
]5)(2[2
yxDD
DDe yx +
−′+−′+= −
)32sin(25)2(4
]5)(2[22
yxDDDD
DDe yx +
−′+′+−′+= −
)32sin(25)]9()6(2)4[(4
]5)(2[yx
DDe yx +
−−+−+−−′+= −
)32sin(125
]5)(2[yx
DDe yx +
−−′+= −
125
)32sin(5)]32[sin(2)]32[sin(2
−+−+′++= − yxyxDyxD
e yx
137
6/4/2015B.Saravanan
)]32sin(5)32cos(6)32cos(4[125
yxyxyxe yx
+−+++−=−
)]32sin(5)32cos(10[125
yxyxe yx
+−+−=−
)]32cos(2)32[sin(25
yxyxe yx
+−+=−
z = C.F + P.I
)]32cos(2)32[sin(25
)()(.).( 21 yxyxe
xyfxyfzeiyx
+−++−++=−
138
6/4/2015B.Saravanan 139
Non-Homogeneous PDE
.hom)1.(
hom),(
)1(),(),(
PDElinearogeneousnoncalledeqnthen
ogeneousnotisDDfpolonomialtheif
yxFzDDfequationtheIn
−′
−−−−−=′
The general solution of equation (1) is z=C.F+P.I
The method to find P.I. is same as those for homogeneousPDE
To find C.F.
0).(..........))((
),(
2211 =−′−−′−−′−′
zcDmDcDmDcDmD
toinDDfFactorize
nn
6/4/2015B.Saravanan 140
)(........)()(. 221121 xmyfexmyfexmyfeFC nn
xcxcxc n ++++++=
0)( 3 =−′− zaDmDsayfactorrepeatedfor
)()()(. 32
21 mxyfexmxyfxemxyfeFCthen axaxax +++++=
6/4/2015B.Saravanan
Problem 1
Solve 0)12)(2( =+′−′− zDDDD
Solution:
The given equation is non-homogeneous.
0)12)(2( =+′−′− zDDDD
)2()2( 210 xyfexyfez xx +++=∴ −
)2()2(.).( 21 xyfexyfzei x +++= −
141
6/4/2015B.Saravanan
Problem 2
Solve 22322 )2()2332( yx eezDDDDDD −+=+′+−′+′−
Solution:
The given equation is non-homogeneous and it can be written as
yxyx eeezDDDD 2346 44)2)(1( −− ++=−′−−′−
)()(.. 22
1 xyfexyfeFC xx +++=
P.I1 = yxe
DDDD06
)2)(1(
1 +
−′−−′−
yxe 06
)206)(106(
1 +
−−−−= xe6
20
1=
142
6/4/2015B.Saravanan
P.I2 = yxe
DDDD404
)2)(1(
1 −
−′−−′−
yxe 40
)240)(140(
14 −
−+−+= ye 4
3
2 −=
P.I3 = yxe
DDDD234
)2)(1(
1 −
−′−−′−
yxe 23
)223)(123(
14 −
−+−+= yxe 23
3
1 −=
z = C.F + P.I1 + P.I2 + P.I3
yxyxxx eeexyfexyfezei 23462
21 3
1
3
2
20
1)()(.).( −− ++++++=
143
6/4/2015B.Saravanan
Problem 3
SolveyxezDDDDDD +=+′++′+′+ 222 )1222(
Solution:
The given equation is non-homogeneous and it can be written as
yxezDDDD +=+′++′+ 2)1)(1(
)()(.. 21 xyfxexyfeFC xx −+−= −−
P.I = yxe
DDDDDD+
+′++′+′+2
22 1222
1
yxe +
+++++= 2
22 1)1(2)2(2)1)(2(2)1()2(
1yxe += 2
16
1
144
6/4/2015B.Saravanan
z = C.F + P.I
yxxx exyfxexyfezei +−− +−+−= 221 16
1)()(.).(
Problem 4
Solve yexzDDDDDD =′++′−′− )362( 22
Solution:
yexzDDDDDDGiven =′++′−′− )362( 22
yexzDDDD =+′−′+ )3)(2(
1,3,2
1,0 2211 =−=−== mcmcHere
145
6/4/2015B.Saravanan
)(2
1.. 2
31
0 xyfexyfeFC xx ++
−= −
)(2
12
31 xyfexyf x ++
−= −
P.I = yexDDDDDD ′++′−′− 362
122
xDDDDDD
e y
)1(36)1()1(2
122 +′+++′−+′−
=
xDDDDDDDD
e y
336122
122 +′++−′−′−−′−
=
146
6/4/2015B.Saravanan
xDDDDDD
e y
′++′−′−+=
2
5212
122
xDDDDDDe y 122
2
521
2
−
′++′−′−+=
xDDDDDDe y
′++′−′−−=2
521
2
22
−= )(2
5
2x
Dx
e y
−=2
5
2x
e y
xDDDDDD
e y
′++′−′−+=
522
122
147
6/4/2015B.Saravanan
z = C.F + P.I
)52(4
)(2
1.).( 2
31 −+++
−= − xe
xyfexyfzeiy
x
)52(4
−= xe y
Problem 5
Solve
Solution:
)2sin()2223( 22 yxyxzDDDDDD +++=′−+′+′−
)2sin()2223( 22 yxyxzDDDDDDGiven +++=′−+′+′−
)2sin()22)(( yxyxzDDDD +++=+′−′−148
6/4/2015B.Saravanan
2,2,1,0 1211 =−=== mcmcHere
)2()(.. 22
10 xyfexyfeFC xx +++= −
)2()( 22
1 xyfexyf x +++= −
)(2223
122
yxDDDDDD
+′−+′+′−
P.I1 =
)()22)((
1yx
DDDD+
+′−′−=
)(
2
2121
1yx
DD
D
DD
+
′−+
′−
=
149
6/4/2015B.Saravanan
)(2
211
2
111
yxDD
D
D
D+
′−+
′−=
−−
)(2
2
2
211
2
12
yxDDDD
D
D
D+
′−+
′−−
′+=
)(42
11
2
1 2
2yxDD
DD
D
D
D
D+
′−+′+−
′+=
)(4242
11
2
12
yxD
D
D
D
DD
D
D
D
D+
′+
′−
′+′−+
′+−=
)(4
3
422
11
2
12
yxD
D
DD
D
D
D+
′−
′++
′+−=
150
6/4/2015B.Saravanan
−−+=2
1
22
1 2 yyxx
−+++−−+=
4
3
24
1
22222
1 22 xxyxyx
x
+′−+′
+
+++′++−+
=
4
)(3)(4
)(
2
)(
2)(
1
2
1
2
yxD
D
yxD
yxD
D
yxDyxyx
D
P.I2 = )2sin(2223
122
yxDDDDDD
+′−+′+′−
151
6/4/2015B.Saravanan
)2sin(22)1(2)2(34
1yx
DD+
′−+−+−−−=
)2sin(22
1yx
DD+
′−=
)2sin()22)(22(
22yx
DDDD
DD +′+′−
′+=
)2sin(44
2222
yxDD
DD +′−′+=
)2sin()1(4)4(4
22yx
DD +−−−′+=
152