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Page 1
Edurev123
3. Equation of 1st Order but Nor of 1st
Degree
3.1. Obtain Clairaut's form of the differential equation
(?? ?? ?? ????
-?? )(?? ????
????
+?? )=?? ?? ????
????
Also find its general solution.
(2011 : 15 Marks)
Solution:
The given differential equation is
(?? ????
????
-?? )(?? ????
????
+?? )=?? 2
????
????
Put
????
????
=?? ,?? 2
=?? ,?? 2
=?? ? ??????? =???? ,??????? =????
?
????
????
=
2?????? 2?????? ??? =
?? ?? ?? ,?? =
????
????
? from (i), (?? ·
?? ?? ?? -?? )(?? ·
?? ?? ?? +?? )=?? 2
?? ?? ?? ? (
?? ?? 2-?? 2
?? )-?? (?? +1)=?? 2
?? ?? ?? ? ???? -?? =
?? 2
?? ?? +1
? ?? =???? -
?? 2
?? ?? +1
? ?? =???? +?? (?? ) which is Clairaut's form.
? Its general solution is
?? =???? +?? (?? ) where ?? is an arbitrary constant
? ?? 2
=?? ?? 2
+?? (?? )
3.2 Solve the DE: ?? =???? -?? ?? where ?? =
????
????
.
(2015 : 13 Marks)
Solution:
Page 2
Edurev123
3. Equation of 1st Order but Nor of 1st
Degree
3.1. Obtain Clairaut's form of the differential equation
(?? ?? ?? ????
-?? )(?? ????
????
+?? )=?? ?? ????
????
Also find its general solution.
(2011 : 15 Marks)
Solution:
The given differential equation is
(?? ????
????
-?? )(?? ????
????
+?? )=?? 2
????
????
Put
????
????
=?? ,?? 2
=?? ,?? 2
=?? ? ??????? =???? ,??????? =????
?
????
????
=
2?????? 2?????? ??? =
?? ?? ?? ,?? =
????
????
? from (i), (?? ·
?? ?? ?? -?? )(?? ·
?? ?? ?? +?? )=?? 2
?? ?? ?? ? (
?? ?? 2-?? 2
?? )-?? (?? +1)=?? 2
?? ?? ?? ? ???? -?? =
?? 2
?? ?? +1
? ?? =???? -
?? 2
?? ?? +1
? ?? =???? +?? (?? ) which is Clairaut's form.
? Its general solution is
?? =???? +?? (?? ) where ?? is an arbitrary constant
? ?? 2
=?? ?? 2
+?? (?? )
3.2 Solve the DE: ?? =???? -?? ?? where ?? =
????
????
.
(2015 : 13 Marks)
Solution:
Differentiate w.r.t. ??
1
?? =?? +?? ????
????
-2?? ????
????
? 1-?? 2
=(?? 1-2?? 2
)
????
????
? (???? -2?? 2
)???? +(?? 2
-1)???? =0 (?????? +?????? =0)
??? ??? =?? ,
??? ??? =2??
1
?? (
??? ??? -
??? ??? )=
-?? ?? 2
-1
I.F. =?? ?
-?? ?? 2
-1
????
=?? -
1
2
log (?? 2
-1)
=
1
v?? 2
-1
D.E. =(
????
v?? 2
-1
-
2?? 2
v?? 2
-1
)???? +(v?? 2
-1)???? =0
?
????
v?? 2
-1
-
2?? 2
v?? 2
-1
= constant
? 2v?? 2
-1-2? ?? ·
?? v?? 2
-1
=?? 1
??? v?? 2
-1-2?? ?
?? v?? 2
-1
+2?
?? v?? 2
-1
=?? ?? ? ?? v?? 2
-1-2?? v?? 2
-1+2? v?? 2
-1=?? 1
??? v?? 2
-1-2?? v?? 2
-1+?? v?? 2
-1-log|(?? +v?? 2
-1|=?? 1
? ?? =?? +
log |?? +v?? 2
-1|+?? 1
v?? 2
-1
?? =???? -?? 2
=?? [log |?? +v?? 2
-1|+?? ]
v?? 2
-1
3.3 Solve the following simultaneous linear ???? (?? +?? )?? =?? +?? ?? and (?? +?? )?? =?? +
?? ?? where ?? and ?? are functions of independent variable ?? and ?? =
?? ????
.
(2017 : 8 Marks)
Solution:
(?? +1)?? =?? +?? ?? (??)
(?? +1)?? =?? +?? ?? (???? )
Applying (?? +1) operator to (i)
(?? +1)
2
?? =(?? +1)?? +(?? +1)?? ??
Page 3
Edurev123
3. Equation of 1st Order but Nor of 1st
Degree
3.1. Obtain Clairaut's form of the differential equation
(?? ?? ?? ????
-?? )(?? ????
????
+?? )=?? ?? ????
????
Also find its general solution.
(2011 : 15 Marks)
Solution:
The given differential equation is
(?? ????
????
-?? )(?? ????
????
+?? )=?? 2
????
????
Put
????
????
=?? ,?? 2
=?? ,?? 2
=?? ? ??????? =???? ,??????? =????
?
????
????
=
2?????? 2?????? ??? =
?? ?? ?? ,?? =
????
????
? from (i), (?? ·
?? ?? ?? -?? )(?? ·
?? ?? ?? +?? )=?? 2
?? ?? ?? ? (
?? ?? 2-?? 2
?? )-?? (?? +1)=?? 2
?? ?? ?? ? ???? -?? =
?? 2
?? ?? +1
? ?? =???? -
?? 2
?? ?? +1
? ?? =???? +?? (?? ) which is Clairaut's form.
? Its general solution is
?? =???? +?? (?? ) where ?? is an arbitrary constant
? ?? 2
=?? ?? 2
+?? (?? )
3.2 Solve the DE: ?? =???? -?? ?? where ?? =
????
????
.
(2015 : 13 Marks)
Solution:
Differentiate w.r.t. ??
1
?? =?? +?? ????
????
-2?? ????
????
? 1-?? 2
=(?? 1-2?? 2
)
????
????
? (???? -2?? 2
)???? +(?? 2
-1)???? =0 (?????? +?????? =0)
??? ??? =?? ,
??? ??? =2??
1
?? (
??? ??? -
??? ??? )=
-?? ?? 2
-1
I.F. =?? ?
-?? ?? 2
-1
????
=?? -
1
2
log (?? 2
-1)
=
1
v?? 2
-1
D.E. =(
????
v?? 2
-1
-
2?? 2
v?? 2
-1
)???? +(v?? 2
-1)???? =0
?
????
v?? 2
-1
-
2?? 2
v?? 2
-1
= constant
? 2v?? 2
-1-2? ?? ·
?? v?? 2
-1
=?? 1
??? v?? 2
-1-2?? ?
?? v?? 2
-1
+2?
?? v?? 2
-1
=?? ?? ? ?? v?? 2
-1-2?? v?? 2
-1+2? v?? 2
-1=?? 1
??? v?? 2
-1-2?? v?? 2
-1+?? v?? 2
-1-log|(?? +v?? 2
-1|=?? 1
? ?? =?? +
log |?? +v?? 2
-1|+?? 1
v?? 2
-1
?? =???? -?? 2
=?? [log |?? +v?? 2
-1|+?? ]
v?? 2
-1
3.3 Solve the following simultaneous linear ???? (?? +?? )?? =?? +?? ?? and (?? +?? )?? =?? +
?? ?? where ?? and ?? are functions of independent variable ?? and ?? =
?? ????
.
(2017 : 8 Marks)
Solution:
(?? +1)?? =?? +?? ?? (??)
(?? +1)?? =?? +?? ?? (???? )
Applying (?? +1) operator to (i)
(?? +1)
2
?? =(?? +1)?? +(?? +1)?? ??
(?? 2
+2?? +1)?? =?? +?? ?? +2?? ?? (?? 2
+2?? )?? =0?? ??
Solution to homogeneous equation
(?? 2
+2?? )?? =0, i.e., ?? (?? +2)?? =0 is
?? ?? =?? 1
+?? 2
?? -2??
Particular integral,
???? =
1
?? (?? +2)
(3?? ?? )=
3
1(1+2)
·?? ?? =?? ??
? ?? =?? 1
+?? 2
?? -2?? +?? ??
From (1)
?? =(?? +1)?? -?? ?? =-2?? 2
?? -2?? +?? ?? +?? 1
+?? 2
?? -2?? ?? =?? 1
-?? 2
?? -2?? +?? ??
3.4 Consider the ???? ,???? ?? ?? -(?? ?? +?? ?? -?? )?? +???? =?? where ?? =
????
????
. Substituting ?? =
?? ?? and ?? =?? ?? reduce the equation to Clairaut's form in terms of ?? ,?? and ?? '
=
????
????
.
Hence or otherwise solve the equation.
(20i7 : 8 Marks)
Solution:
???? ?? 2
-(?? 2
+?? 2
-1)?? +???? =0
?? =?? 2
? ???? =2?????? ?? =?? 2
????? =2?????? ?
????
????
=
?? ?? ????
????
i.e., ?? '
=
?? ?? ?? or ?? =
?? ?? ?? '
(??)
Substituting in (i),
???? ·
?? 2
?? 2
·?? 2
-(?? +?? -1)
?? ?? ?? '
+???? =0
? ?? 2
·?? 2
-(?? +?? -1)?? '
+?? 2
=0
?? 2
?? '2
-(?? +?? -1)?? '
+?? =0
(?? '
-1)?? =?? ?? '
(?? '
-1)+?? '
?? =?? ?? '
+
?? '
?? '
-1
Page 4
Edurev123
3. Equation of 1st Order but Nor of 1st
Degree
3.1. Obtain Clairaut's form of the differential equation
(?? ?? ?? ????
-?? )(?? ????
????
+?? )=?? ?? ????
????
Also find its general solution.
(2011 : 15 Marks)
Solution:
The given differential equation is
(?? ????
????
-?? )(?? ????
????
+?? )=?? 2
????
????
Put
????
????
=?? ,?? 2
=?? ,?? 2
=?? ? ??????? =???? ,??????? =????
?
????
????
=
2?????? 2?????? ??? =
?? ?? ?? ,?? =
????
????
? from (i), (?? ·
?? ?? ?? -?? )(?? ·
?? ?? ?? +?? )=?? 2
?? ?? ?? ? (
?? ?? 2-?? 2
?? )-?? (?? +1)=?? 2
?? ?? ?? ? ???? -?? =
?? 2
?? ?? +1
? ?? =???? -
?? 2
?? ?? +1
? ?? =???? +?? (?? ) which is Clairaut's form.
? Its general solution is
?? =???? +?? (?? ) where ?? is an arbitrary constant
? ?? 2
=?? ?? 2
+?? (?? )
3.2 Solve the DE: ?? =???? -?? ?? where ?? =
????
????
.
(2015 : 13 Marks)
Solution:
Differentiate w.r.t. ??
1
?? =?? +?? ????
????
-2?? ????
????
? 1-?? 2
=(?? 1-2?? 2
)
????
????
? (???? -2?? 2
)???? +(?? 2
-1)???? =0 (?????? +?????? =0)
??? ??? =?? ,
??? ??? =2??
1
?? (
??? ??? -
??? ??? )=
-?? ?? 2
-1
I.F. =?? ?
-?? ?? 2
-1
????
=?? -
1
2
log (?? 2
-1)
=
1
v?? 2
-1
D.E. =(
????
v?? 2
-1
-
2?? 2
v?? 2
-1
)???? +(v?? 2
-1)???? =0
?
????
v?? 2
-1
-
2?? 2
v?? 2
-1
= constant
? 2v?? 2
-1-2? ?? ·
?? v?? 2
-1
=?? 1
??? v?? 2
-1-2?? ?
?? v?? 2
-1
+2?
?? v?? 2
-1
=?? ?? ? ?? v?? 2
-1-2?? v?? 2
-1+2? v?? 2
-1=?? 1
??? v?? 2
-1-2?? v?? 2
-1+?? v?? 2
-1-log|(?? +v?? 2
-1|=?? 1
? ?? =?? +
log |?? +v?? 2
-1|+?? 1
v?? 2
-1
?? =???? -?? 2
=?? [log |?? +v?? 2
-1|+?? ]
v?? 2
-1
3.3 Solve the following simultaneous linear ???? (?? +?? )?? =?? +?? ?? and (?? +?? )?? =?? +
?? ?? where ?? and ?? are functions of independent variable ?? and ?? =
?? ????
.
(2017 : 8 Marks)
Solution:
(?? +1)?? =?? +?? ?? (??)
(?? +1)?? =?? +?? ?? (???? )
Applying (?? +1) operator to (i)
(?? +1)
2
?? =(?? +1)?? +(?? +1)?? ??
(?? 2
+2?? +1)?? =?? +?? ?? +2?? ?? (?? 2
+2?? )?? =0?? ??
Solution to homogeneous equation
(?? 2
+2?? )?? =0, i.e., ?? (?? +2)?? =0 is
?? ?? =?? 1
+?? 2
?? -2??
Particular integral,
???? =
1
?? (?? +2)
(3?? ?? )=
3
1(1+2)
·?? ?? =?? ??
? ?? =?? 1
+?? 2
?? -2?? +?? ??
From (1)
?? =(?? +1)?? -?? ?? =-2?? 2
?? -2?? +?? ?? +?? 1
+?? 2
?? -2?? ?? =?? 1
-?? 2
?? -2?? +?? ??
3.4 Consider the ???? ,???? ?? ?? -(?? ?? +?? ?? -?? )?? +???? =?? where ?? =
????
????
. Substituting ?? =
?? ?? and ?? =?? ?? reduce the equation to Clairaut's form in terms of ?? ,?? and ?? '
=
????
????
.
Hence or otherwise solve the equation.
(20i7 : 8 Marks)
Solution:
???? ?? 2
-(?? 2
+?? 2
-1)?? +???? =0
?? =?? 2
? ???? =2?????? ?? =?? 2
????? =2?????? ?
????
????
=
?? ?? ????
????
i.e., ?? '
=
?? ?? ?? or ?? =
?? ?? ?? '
(??)
Substituting in (i),
???? ·
?? 2
?? 2
·?? 2
-(?? +?? -1)
?? ?? ?? '
+???? =0
? ?? 2
·?? 2
-(?? +?? -1)?? '
+?? 2
=0
?? 2
?? '2
-(?? +?? -1)?? '
+?? =0
(?? '
-1)?? =?? ?? '
(?? '
-1)+?? '
?? =?? ?? '
+
?? '
?? '
-1
This is Clairaut's form
? ?? =???? +
?? ?? -1
Solution to (1)? ?? 2
=?? ?? 2
+
?? ?? -1
3.5 Solve : (
????
????
)
?? ?? +?? ????
????
?? -?? =?? .
(2018: 13 marks)
Solution:
Given equation is
(
????
????
)
2
?? +2
????
????
?? -?? =0
Let
????
?? ?? =??
? equation becomes ?? ?? 2
+2???? -?? =0?2???? =?? -?? ?? 2
? ?? =
?? 2?? -
????
2
?
????
????
=
1
?? =
1
2?? -
?? 2?? 2
????
????
·
?? 2
-
?? 2
????
????
?
1
2?? =
-?? 2
-
?????? 2????
(
1
?? 2
+1)
?
1
2
(?? 2
+1)
?? =
-?? 2
????
????
(?? 2
+1)
?? 2
?
?? ????
=
-????
??
Integrating on both sides, we get
log ?? =log ?? +log ?? , where ?? is a constant.
? ???? =?? ??? =
?? ??
Putting this value in given differential equation, we get
Page 5
Edurev123
3. Equation of 1st Order but Nor of 1st
Degree
3.1. Obtain Clairaut's form of the differential equation
(?? ?? ?? ????
-?? )(?? ????
????
+?? )=?? ?? ????
????
Also find its general solution.
(2011 : 15 Marks)
Solution:
The given differential equation is
(?? ????
????
-?? )(?? ????
????
+?? )=?? 2
????
????
Put
????
????
=?? ,?? 2
=?? ,?? 2
=?? ? ??????? =???? ,??????? =????
?
????
????
=
2?????? 2?????? ??? =
?? ?? ?? ,?? =
????
????
? from (i), (?? ·
?? ?? ?? -?? )(?? ·
?? ?? ?? +?? )=?? 2
?? ?? ?? ? (
?? ?? 2-?? 2
?? )-?? (?? +1)=?? 2
?? ?? ?? ? ???? -?? =
?? 2
?? ?? +1
? ?? =???? -
?? 2
?? ?? +1
? ?? =???? +?? (?? ) which is Clairaut's form.
? Its general solution is
?? =???? +?? (?? ) where ?? is an arbitrary constant
? ?? 2
=?? ?? 2
+?? (?? )
3.2 Solve the DE: ?? =???? -?? ?? where ?? =
????
????
.
(2015 : 13 Marks)
Solution:
Differentiate w.r.t. ??
1
?? =?? +?? ????
????
-2?? ????
????
? 1-?? 2
=(?? 1-2?? 2
)
????
????
? (???? -2?? 2
)???? +(?? 2
-1)???? =0 (?????? +?????? =0)
??? ??? =?? ,
??? ??? =2??
1
?? (
??? ??? -
??? ??? )=
-?? ?? 2
-1
I.F. =?? ?
-?? ?? 2
-1
????
=?? -
1
2
log (?? 2
-1)
=
1
v?? 2
-1
D.E. =(
????
v?? 2
-1
-
2?? 2
v?? 2
-1
)???? +(v?? 2
-1)???? =0
?
????
v?? 2
-1
-
2?? 2
v?? 2
-1
= constant
? 2v?? 2
-1-2? ?? ·
?? v?? 2
-1
=?? 1
??? v?? 2
-1-2?? ?
?? v?? 2
-1
+2?
?? v?? 2
-1
=?? ?? ? ?? v?? 2
-1-2?? v?? 2
-1+2? v?? 2
-1=?? 1
??? v?? 2
-1-2?? v?? 2
-1+?? v?? 2
-1-log|(?? +v?? 2
-1|=?? 1
? ?? =?? +
log |?? +v?? 2
-1|+?? 1
v?? 2
-1
?? =???? -?? 2
=?? [log |?? +v?? 2
-1|+?? ]
v?? 2
-1
3.3 Solve the following simultaneous linear ???? (?? +?? )?? =?? +?? ?? and (?? +?? )?? =?? +
?? ?? where ?? and ?? are functions of independent variable ?? and ?? =
?? ????
.
(2017 : 8 Marks)
Solution:
(?? +1)?? =?? +?? ?? (??)
(?? +1)?? =?? +?? ?? (???? )
Applying (?? +1) operator to (i)
(?? +1)
2
?? =(?? +1)?? +(?? +1)?? ??
(?? 2
+2?? +1)?? =?? +?? ?? +2?? ?? (?? 2
+2?? )?? =0?? ??
Solution to homogeneous equation
(?? 2
+2?? )?? =0, i.e., ?? (?? +2)?? =0 is
?? ?? =?? 1
+?? 2
?? -2??
Particular integral,
???? =
1
?? (?? +2)
(3?? ?? )=
3
1(1+2)
·?? ?? =?? ??
? ?? =?? 1
+?? 2
?? -2?? +?? ??
From (1)
?? =(?? +1)?? -?? ?? =-2?? 2
?? -2?? +?? ?? +?? 1
+?? 2
?? -2?? ?? =?? 1
-?? 2
?? -2?? +?? ??
3.4 Consider the ???? ,???? ?? ?? -(?? ?? +?? ?? -?? )?? +???? =?? where ?? =
????
????
. Substituting ?? =
?? ?? and ?? =?? ?? reduce the equation to Clairaut's form in terms of ?? ,?? and ?? '
=
????
????
.
Hence or otherwise solve the equation.
(20i7 : 8 Marks)
Solution:
???? ?? 2
-(?? 2
+?? 2
-1)?? +???? =0
?? =?? 2
? ???? =2?????? ?? =?? 2
????? =2?????? ?
????
????
=
?? ?? ????
????
i.e., ?? '
=
?? ?? ?? or ?? =
?? ?? ?? '
(??)
Substituting in (i),
???? ·
?? 2
?? 2
·?? 2
-(?? +?? -1)
?? ?? ?? '
+???? =0
? ?? 2
·?? 2
-(?? +?? -1)?? '
+?? 2
=0
?? 2
?? '2
-(?? +?? -1)?? '
+?? =0
(?? '
-1)?? =?? ?? '
(?? '
-1)+?? '
?? =?? ?? '
+
?? '
?? '
-1
This is Clairaut's form
? ?? =???? +
?? ?? -1
Solution to (1)? ?? 2
=?? ?? 2
+
?? ?? -1
3.5 Solve : (
????
????
)
?? ?? +?? ????
????
?? -?? =?? .
(2018: 13 marks)
Solution:
Given equation is
(
????
????
)
2
?? +2
????
????
?? -?? =0
Let
????
?? ?? =??
? equation becomes ?? ?? 2
+2???? -?? =0?2???? =?? -?? ?? 2
? ?? =
?? 2?? -
????
2
?
????
????
=
1
?? =
1
2?? -
?? 2?? 2
????
????
·
?? 2
-
?? 2
????
????
?
1
2?? =
-?? 2
-
?????? 2????
(
1
?? 2
+1)
?
1
2
(?? 2
+1)
?? =
-?? 2
????
????
(?? 2
+1)
?? 2
?
?? ????
=
-????
??
Integrating on both sides, we get
log ?? =log ?? +log ?? , where ?? is a constant.
? ???? =?? ??? =
?? ??
Putting this value in given differential equation, we get
2·
?? ?? ?? =?? -?? ?? 2
?? ?? ?
2????
?? =?? -
?? 2
?? ?2???? =?? 2
-?? 2
? ?? 2
-2???? =?? 2
3.6 Obtain the singular solution of the differential equation
(
????
????
)
?? (
?? ?? )
?? ?????? ?? ?? -?? (
????
????
)(
?? ?? )+(
?? ?? )
?? ?????????? ?? ?? =??
Also, find the complete primitive of the given differential equation. Give the
geometrical interpretations of the complete primitive and singular solution.
Solution:
Given :
(
????
????
)
2
(
?? ?? )
2
cot
2
?? -2(
????
????
)(
?? ?? )+(
?? ?? )
2
cosec
2
?? =1
which can be written as :
?? 2
?? 2
cot
2
?? -2?????? +?? 2
cosec
2
?? =?? 2
?? 2
?? 2
cos
2
?? sin
2
?? -2?????? +?? 2
1
sin
2
?? =?? 2
? ?? 2
?? 2
cos
2
?? -2?????? sin
2
?? +?? 2
-?? 2
sin
2
?? =0
? (???? )
2
-(2???? )?? tan
2
?? +(?? 2
sec
2
?? -?? 2
tan
2
?? )=0
???? =
2?? tan
2
?? +v4?? 2
tan
4
?? -4(?? 2
sec
2
?? -?? 2
tan
2
?? )
2
???? =?? tan
2
?? ±v?? 2
tan
2
?? (tan
2
?? +1)-?? 2
sec
2
?? ???? =?? tan
2
?? ±sec ?? v?? 2
tan
2
?? -?? 2
?????? -?? tan
2
?????? =±sec ?? v(?? 2
tar
2
?? -?? 2
)????
? ±
?? tan
2
?????? -???? ?? v(?? 2
tan
2
?? -?? 2
)
=-sec ?????? ?????????????????????? , ±v?? 2
tan
2
?? -?? 2
=?? -?? sec ?? ???????????????? , ?? 2
+tan
2
?? -?? 2
=?? 2
-2???? sec ?? +?? 2
sec
2
?? =?? 2
(tan
2
?? -sec
2
?? )-?? 2
=?? 2
-2cosec ?? =-?? 2
-?? 2
=?? 2
=2???? sec ?? [?-1=tan
2
-sec
2
?? ]
? ?? 2
+?? 2
-2???? sec ?? +?? 2
=0
From (?? )
?? 2
?? 2
cos
2
?? -2?????? sin
2
?? +?? 2
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FAQs on Equation of 1st Order but Nor of 1st Degree - Mathematics Optional Notes for UPSC
| 1. What is the difference between an equation of 1st order and an equation of 1st degree? |  |
Ans. An equation of 1st order refers to an equation where the highest derivative present is the first derivative, while an equation of 1st degree refers to an equation where the highest power of the unknown variable is 1.
| 2. How can we solve an equation of 1st order but not of 1st degree? | |
Ans. To solve an equation of 1st order but not of 1st degree, we typically use methods such as separation of variables, integrating factors, or substitution to transform the equation into a form that can be easily solved.
| 3. Can you provide an example of an equation of 1st order but not of 1st degree? | |
Ans. One example of an equation of 1st order but not of 1st degree is the Bernoulli differential equation, which can be written in the form dy/dx = P(x)y + Q(x)y^n, where n is a constant other than 1.
| 4. What are some real-world applications of equations of 1st order but not of 1st degree? | |
Ans. Equations of 1st order but not of 1st degree are commonly used in various fields such as physics, biology, economics, and engineering to model dynamic systems, population growth, chemical reactions, and more.
| 5. How important are equations of 1st order but not of 1st degree in the field of mathematics? | |
Ans. Equations of 1st order but not of 1st degree are fundamental in the study of differential equations and have wide-ranging applications in various branches of mathematics and science, making them a crucial topic for students and researchers alike.
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Dec 22, 2024 Last updated |
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