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Page 1
Edurev123
2. Solution of Quasilinear Partial
Differential Equations of 1st Order,
Lagrange's Auxiliary Equations and
Charpit's Auxiliary Equations
2.1. Find the integral surface of :
?? ?? ?? +?? ?? ?? +?? ?? =??
which passes through the curve:
???? =?? +?? ,?? =??
(2009 : 6 Marks)
Solution:
Given ??? 2
?? +?? 2
?? +?? 2
=0 or ?? 2
?? +?? 2
?? =?? 2
(1)
Given curve is
???? =?? +?? ,?? =1 (2)
Here Lagrange's auxiliary equations are
????
?? 2
=
????
?? 2
=
????
(-?? 2
)
(3)
Taking the first and third fractions,
?? 2
???? +?? 2
???? =0
?????????????? ?? ?????? ???????????????????????????????????-(
1
?? )-(
1
?? )=-?? 1
or ?
1
?? +
1
?? =?? 1
?????????????????????????????????????????????????????????????????????????????????????(4)
?? 2
???? +?? 2
???? =0
(
1
?? )+(
1
?? )=-?? 2
or
1
?? +
1
?? =?? 2
(5)
Adding (4) and (5)
Page 2
Edurev123
2. Solution of Quasilinear Partial
Differential Equations of 1st Order,
Lagrange's Auxiliary Equations and
Charpit's Auxiliary Equations
2.1. Find the integral surface of :
?? ?? ?? +?? ?? ?? +?? ?? =??
which passes through the curve:
???? =?? +?? ,?? =??
(2009 : 6 Marks)
Solution:
Given ??? 2
?? +?? 2
?? +?? 2
=0 or ?? 2
?? +?? 2
?? =?? 2
(1)
Given curve is
???? =?? +?? ,?? =1 (2)
Here Lagrange's auxiliary equations are
????
?? 2
=
????
?? 2
=
????
(-?? 2
)
(3)
Taking the first and third fractions,
?? 2
???? +?? 2
???? =0
?????????????? ?? ?????? ???????????????????????????????????-(
1
?? )-(
1
?? )=-?? 1
or ?
1
?? +
1
?? =?? 1
?????????????????????????????????????????????????????????????????????????????????????(4)
?? 2
???? +?? 2
???? =0
(
1
?? )+(
1
?? )=-?? 2
or
1
?? +
1
?? =?? 2
(5)
Adding (4) and (5)
1
?? +
1
?? +
2
?? =?? 1
+?? 2
?? +?? ????
+
2
?? =?? 1
+?? 2
(?? +?? )(???? )+2?=?? 1
+?? 2
, using (2)
?? 1
+?? 2
=3
Substituting the values of ?? 1
and ?? 2
from (4) and (5) in (6), we get
1
?? +
1
?? +
1
?? +
1
?? =3 or ???? +2???? +???? =3??????
2.2 Solve the PDE :
(?? +?? ?? )
?? ?? ?? ?? +(?? ???? -?? )
?? ?? ?? ?? =?? ?? ?? +??
(2011: 12 marks)
Solution:
The given partial differential equation is
(?? +2?? )
??? ??? +(4???? -?? )
??? ??? =2?? 2
+??
Putting
??? ??? =?? and
??? ??? =?? ,
(?? +2?? )?? +(4???? -?? )?? =2?? 2
+??
It is of the form of
?? ?? +?? ?? =??
The Lagrange's auxiliary equations are
????
?? =
????
?? =
????
?? or
????
?? +2?? =
????
4???? -?? =
????
2?? 2
+??
Using 2?? ,-1,-1 as multipliers, each fraction in (i) is equal to
2?????? -???? -????
2?? 2
+4???? -4???? +?? -2?? 2
-?? ?=
2?????? -???? -????
0
???????????????????????????????????????????????????????????????????????????? 2
-?? -?? ?=?? 1
Page 3
Edurev123
2. Solution of Quasilinear Partial
Differential Equations of 1st Order,
Lagrange's Auxiliary Equations and
Charpit's Auxiliary Equations
2.1. Find the integral surface of :
?? ?? ?? +?? ?? ?? +?? ?? =??
which passes through the curve:
???? =?? +?? ,?? =??
(2009 : 6 Marks)
Solution:
Given ??? 2
?? +?? 2
?? +?? 2
=0 or ?? 2
?? +?? 2
?? =?? 2
(1)
Given curve is
???? =?? +?? ,?? =1 (2)
Here Lagrange's auxiliary equations are
????
?? 2
=
????
?? 2
=
????
(-?? 2
)
(3)
Taking the first and third fractions,
?? 2
???? +?? 2
???? =0
?????????????? ?? ?????? ???????????????????????????????????-(
1
?? )-(
1
?? )=-?? 1
or ?
1
?? +
1
?? =?? 1
?????????????????????????????????????????????????????????????????????????????????????(4)
?? 2
???? +?? 2
???? =0
(
1
?? )+(
1
?? )=-?? 2
or
1
?? +
1
?? =?? 2
(5)
Adding (4) and (5)
1
?? +
1
?? +
2
?? =?? 1
+?? 2
?? +?? ????
+
2
?? =?? 1
+?? 2
(?? +?? )(???? )+2?=?? 1
+?? 2
, using (2)
?? 1
+?? 2
=3
Substituting the values of ?? 1
and ?? 2
from (4) and (5) in (6), we get
1
?? +
1
?? +
1
?? +
1
?? =3 or ???? +2???? +???? =3??????
2.2 Solve the PDE :
(?? +?? ?? )
?? ?? ?? ?? +(?? ???? -?? )
?? ?? ?? ?? =?? ?? ?? +??
(2011: 12 marks)
Solution:
The given partial differential equation is
(?? +2?? )
??? ??? +(4???? -?? )
??? ??? =2?? 2
+??
Putting
??? ??? =?? and
??? ??? =?? ,
(?? +2?? )?? +(4???? -?? )?? =2?? 2
+??
It is of the form of
?? ?? +?? ?? =??
The Lagrange's auxiliary equations are
????
?? =
????
?? =
????
?? or
????
?? +2?? =
????
4???? -?? =
????
2?? 2
+??
Using 2?? ,-1,-1 as multipliers, each fraction in (i) is equal to
2?????? -???? -????
2?? 2
+4???? -4???? +?? -2?? 2
-?? ?=
2?????? -???? -????
0
???????????????????????????????????????????????????????????????????????????? 2
-?? -?? ?=?? 1
Taking ?? ,?? ,-2?? as multipliers, we have,
?????? +?????? -2??????
???? +2???? +4?? ?? 2
-???? -4?? 2
?? -2????
=
?????? +?????? -2??????
0
??????? -?? 2
=?? 2
????? (?? 2
-?? -?? ,???? -?? 2
)=0
2.3 Solve the partial differential equation ???? +???? =?? ?? .
(2012 : 20 Marks)
Solution:
The given equation is
???? +???? =32 (??)
The Lagrange's Auxiliary equation is
????
?? =
????
?? =
????
?? (???? )
Taking first two fractions of equation (ii),
????
?? =
????
??
on?integrating,?????????????????????????????????????????????????log??? ?=log??? +log??? 1
?? ?=?? ?? 1
?
?? ?? =?? 1
Taking last two fractions of equation (ii),
????????????????????????????????????????????
????
?? =
????
3?? ?3log??? =log??? +log??? 2
?????????????????????????????????????????
?? 3
?? =?? 2
So, required general solution of (i) is
?? (?? 1
,?? 2
)?=0
?? (
?? ?? ,
?? 3
?? )?=0
2.4 Solve the partial differential equation
(?? ?? +?? ?? -?? ?? )?? -?? ?????? +?? ???? =??
where
Page 4
Edurev123
2. Solution of Quasilinear Partial
Differential Equations of 1st Order,
Lagrange's Auxiliary Equations and
Charpit's Auxiliary Equations
2.1. Find the integral surface of :
?? ?? ?? +?? ?? ?? +?? ?? =??
which passes through the curve:
???? =?? +?? ,?? =??
(2009 : 6 Marks)
Solution:
Given ??? 2
?? +?? 2
?? +?? 2
=0 or ?? 2
?? +?? 2
?? =?? 2
(1)
Given curve is
???? =?? +?? ,?? =1 (2)
Here Lagrange's auxiliary equations are
????
?? 2
=
????
?? 2
=
????
(-?? 2
)
(3)
Taking the first and third fractions,
?? 2
???? +?? 2
???? =0
?????????????? ?? ?????? ???????????????????????????????????-(
1
?? )-(
1
?? )=-?? 1
or ?
1
?? +
1
?? =?? 1
?????????????????????????????????????????????????????????????????????????????????????(4)
?? 2
???? +?? 2
???? =0
(
1
?? )+(
1
?? )=-?? 2
or
1
?? +
1
?? =?? 2
(5)
Adding (4) and (5)
1
?? +
1
?? +
2
?? =?? 1
+?? 2
?? +?? ????
+
2
?? =?? 1
+?? 2
(?? +?? )(???? )+2?=?? 1
+?? 2
, using (2)
?? 1
+?? 2
=3
Substituting the values of ?? 1
and ?? 2
from (4) and (5) in (6), we get
1
?? +
1
?? +
1
?? +
1
?? =3 or ???? +2???? +???? =3??????
2.2 Solve the PDE :
(?? +?? ?? )
?? ?? ?? ?? +(?? ???? -?? )
?? ?? ?? ?? =?? ?? ?? +??
(2011: 12 marks)
Solution:
The given partial differential equation is
(?? +2?? )
??? ??? +(4???? -?? )
??? ??? =2?? 2
+??
Putting
??? ??? =?? and
??? ??? =?? ,
(?? +2?? )?? +(4???? -?? )?? =2?? 2
+??
It is of the form of
?? ?? +?? ?? =??
The Lagrange's auxiliary equations are
????
?? =
????
?? =
????
?? or
????
?? +2?? =
????
4???? -?? =
????
2?? 2
+??
Using 2?? ,-1,-1 as multipliers, each fraction in (i) is equal to
2?????? -???? -????
2?? 2
+4???? -4???? +?? -2?? 2
-?? ?=
2?????? -???? -????
0
???????????????????????????????????????????????????????????????????????????? 2
-?? -?? ?=?? 1
Taking ?? ,?? ,-2?? as multipliers, we have,
?????? +?????? -2??????
???? +2???? +4?? ?? 2
-???? -4?? 2
?? -2????
=
?????? +?????? -2??????
0
??????? -?? 2
=?? 2
????? (?? 2
-?? -?? ,???? -?? 2
)=0
2.3 Solve the partial differential equation ???? +???? =?? ?? .
(2012 : 20 Marks)
Solution:
The given equation is
???? +???? =32 (??)
The Lagrange's Auxiliary equation is
????
?? =
????
?? =
????
?? (???? )
Taking first two fractions of equation (ii),
????
?? =
????
??
on?integrating,?????????????????????????????????????????????????log??? ?=log??? +log??? 1
?? ?=?? ?? 1
?
?? ?? =?? 1
Taking last two fractions of equation (ii),
????????????????????????????????????????????
????
?? =
????
3?? ?3log??? =log??? +log??? 2
?????????????????????????????????????????
?? 3
?? =?? 2
So, required general solution of (i) is
?? (?? 1
,?? 2
)?=0
?? (
?? ?? ,
?? 3
?? )?=0
2.4 Solve the partial differential equation
(?? ?? +?? ?? -?? ?? )?? -?? ?????? +?? ???? =??
where
?? =
?? ?? ?? ?? ,?? =
?? ?? ?? ??
(2015 : 10 Marks)
Solution:
Given equation is :
(?? 2
+?? 2
-?? 2
)?? -2?????? +2???? =0
(?? 2
+?? 2
-?? 2
)?? -2?????? =-2????
Lagrange's auxiliary equation are :
????
?? 2
+?? 2
-?? 2
=
????
-2????
=
????
-2????
Consider
????
-2????
=
????
-2×?? ?
????
?? =
????
??
Integrating it,
? log??? =log??? +log?(?? ), where ?? is a constant:
? log?(
?? ?? ) =log??? ? ?? =????
(??)
Also, using ?? ,?? and ?? as multipliers, we get
???
?????? +?????? +??????
?? (?? 2
+?? 2
-?? 2
)-?? (2?? 2
)-?? (2?? 2
)
=
????
-2????
????????????????
?????? +?????? +??????
?? (?? 2
+?? 2
-?? 2
-2?? 2
-2?? 2
)
=
????
????
???????????????????????????????
2?????? +2?????? +2??????
?? 2
+?? 2
+?? 2
=
????
??
Integrating both sides, we get
log?(?? 2
+?? 2
+?? 2
)=log??? +log??? ; where ?? is a constant.
???? 2
+?? 2
+?? 2
= by (???? )
From (i) and (ii), solution is given by
?? (?? -???? ,?? 2
+?? 2
+?? 2
-???? )=0
Page 5
Edurev123
2. Solution of Quasilinear Partial
Differential Equations of 1st Order,
Lagrange's Auxiliary Equations and
Charpit's Auxiliary Equations
2.1. Find the integral surface of :
?? ?? ?? +?? ?? ?? +?? ?? =??
which passes through the curve:
???? =?? +?? ,?? =??
(2009 : 6 Marks)
Solution:
Given ??? 2
?? +?? 2
?? +?? 2
=0 or ?? 2
?? +?? 2
?? =?? 2
(1)
Given curve is
???? =?? +?? ,?? =1 (2)
Here Lagrange's auxiliary equations are
????
?? 2
=
????
?? 2
=
????
(-?? 2
)
(3)
Taking the first and third fractions,
?? 2
???? +?? 2
???? =0
?????????????? ?? ?????? ???????????????????????????????????-(
1
?? )-(
1
?? )=-?? 1
or ?
1
?? +
1
?? =?? 1
?????????????????????????????????????????????????????????????????????????????????????(4)
?? 2
???? +?? 2
???? =0
(
1
?? )+(
1
?? )=-?? 2
or
1
?? +
1
?? =?? 2
(5)
Adding (4) and (5)
1
?? +
1
?? +
2
?? =?? 1
+?? 2
?? +?? ????
+
2
?? =?? 1
+?? 2
(?? +?? )(???? )+2?=?? 1
+?? 2
, using (2)
?? 1
+?? 2
=3
Substituting the values of ?? 1
and ?? 2
from (4) and (5) in (6), we get
1
?? +
1
?? +
1
?? +
1
?? =3 or ???? +2???? +???? =3??????
2.2 Solve the PDE :
(?? +?? ?? )
?? ?? ?? ?? +(?? ???? -?? )
?? ?? ?? ?? =?? ?? ?? +??
(2011: 12 marks)
Solution:
The given partial differential equation is
(?? +2?? )
??? ??? +(4???? -?? )
??? ??? =2?? 2
+??
Putting
??? ??? =?? and
??? ??? =?? ,
(?? +2?? )?? +(4???? -?? )?? =2?? 2
+??
It is of the form of
?? ?? +?? ?? =??
The Lagrange's auxiliary equations are
????
?? =
????
?? =
????
?? or
????
?? +2?? =
????
4???? -?? =
????
2?? 2
+??
Using 2?? ,-1,-1 as multipliers, each fraction in (i) is equal to
2?????? -???? -????
2?? 2
+4???? -4???? +?? -2?? 2
-?? ?=
2?????? -???? -????
0
???????????????????????????????????????????????????????????????????????????? 2
-?? -?? ?=?? 1
Taking ?? ,?? ,-2?? as multipliers, we have,
?????? +?????? -2??????
???? +2???? +4?? ?? 2
-???? -4?? 2
?? -2????
=
?????? +?????? -2??????
0
??????? -?? 2
=?? 2
????? (?? 2
-?? -?? ,???? -?? 2
)=0
2.3 Solve the partial differential equation ???? +???? =?? ?? .
(2012 : 20 Marks)
Solution:
The given equation is
???? +???? =32 (??)
The Lagrange's Auxiliary equation is
????
?? =
????
?? =
????
?? (???? )
Taking first two fractions of equation (ii),
????
?? =
????
??
on?integrating,?????????????????????????????????????????????????log??? ?=log??? +log??? 1
?? ?=?? ?? 1
?
?? ?? =?? 1
Taking last two fractions of equation (ii),
????????????????????????????????????????????
????
?? =
????
3?? ?3log??? =log??? +log??? 2
?????????????????????????????????????????
?? 3
?? =?? 2
So, required general solution of (i) is
?? (?? 1
,?? 2
)?=0
?? (
?? ?? ,
?? 3
?? )?=0
2.4 Solve the partial differential equation
(?? ?? +?? ?? -?? ?? )?? -?? ?????? +?? ???? =??
where
?? =
?? ?? ?? ?? ,?? =
?? ?? ?? ??
(2015 : 10 Marks)
Solution:
Given equation is :
(?? 2
+?? 2
-?? 2
)?? -2?????? +2???? =0
(?? 2
+?? 2
-?? 2
)?? -2?????? =-2????
Lagrange's auxiliary equation are :
????
?? 2
+?? 2
-?? 2
=
????
-2????
=
????
-2????
Consider
????
-2????
=
????
-2×?? ?
????
?? =
????
??
Integrating it,
? log??? =log??? +log?(?? ), where ?? is a constant:
? log?(
?? ?? ) =log??? ? ?? =????
(??)
Also, using ?? ,?? and ?? as multipliers, we get
???
?????? +?????? +??????
?? (?? 2
+?? 2
-?? 2
)-?? (2?? 2
)-?? (2?? 2
)
=
????
-2????
????????????????
?????? +?????? +??????
?? (?? 2
+?? 2
-?? 2
-2?? 2
-2?? 2
)
=
????
????
???????????????????????????????
2?????? +2?????? +2??????
?? 2
+?? 2
+?? 2
=
????
??
Integrating both sides, we get
log?(?? 2
+?? 2
+?? 2
)=log??? +log??? ; where ?? is a constant.
???? 2
+?? 2
+?? 2
= by (???? )
From (i) and (ii), solution is given by
?? (?? -???? ,?? 2
+?? 2
+?? 2
-???? )=0
2.5 Solve for the general solution ?? ?????? ?(?? +?? )+?? ?????? ?(?? +?? )=?? , where ?? =
?? ?? ?? ?? and
?? =
?? ?? ?? ?? .
(2015 : 15 Marks)
Solution:
Given equation is :
?? cos?(?? +?? )+?? sin?(?? +?? )=??
We first write Lagrange's auxiliary equation for given equation
i.e.,
????
cos?(?? +?? )
=
????
sin?(?? +?? )
=
????
??
Consider
???? +????
cos?(?? +?? )+sin?(?? +?? )
=
????
??
?
???? +????
v2×
1
v2
{cos?(?? +?? )+sin?(?? +?? )}
=
????
??
?
???? +????
v2{
cos?(?? +?? )
v2
+
sin?(?? +?? )
v2
}
=
????
??
?
???? +????
v2{cos?(?? +?? )sin?(
?? 4
)+sin?(?? +?? )cos?(
?? 4
)}
=
????
??
?
?? (?? +?? +
?? 4
)
v2sin?(?? +?? +
?? 4
)
=
????
?? ??cosec(?? +?? +
?? 4
)?? (?? +?? +
?? 4
)=
????
??
Integrating both sides, we get
??????????????????????????-log?[cosec?(?? +?? +
?? 4
)+cot?(?? +?? +
?? 4
)]=
????
?? log??? +log??? ;
where ?? is a constant.
?log??? +log?{cosec?(?? +?? +
?? 4
)+cot?(?? +?? +
?? 4
)}+log??? =0
? ????????????????????????????????????????? (cosec?(?? +?? +
?? 4
)+cot?(?? +?? +
?? 4
))=
1
??
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Solution of Quasilinear Partial Differential Equations of 1st Order, Lagrange's Auxiliary Equations UPSC Questions
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