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Page 1
Edurev123
2. Equation of 1st Order and 1st Degree
2.1 Solve :
????
????
=
?? ?? (?? -?? )
?? ?? ?? ?? -?? ?? ?? -?? ?? ?? ,?? (?? )=??
(2009 : 20 Marks)
Solution:
Approach : We check for exactness and find it to be so.
????
????
=
?? 2
(?? -?? )
3?? ?? 2
-?? 2
?? -4?? 3
??? 2
(?? -?? )???? +(3?? ?? 2
-?? 2
?? -4?? 3
)???? =0
Comparing to ?????? +?????? =0
?? =?? 2
(?? -?? )
?? =3?? ?? 2
-?? 2
?? -4?? 3
??? ??? =3?? 2
-2????
??? ??? =3?? 2
-2????
?
??? ??? =
??? ??? , so the equation is exact.
Solution of an exact equation :
??? (treating ?? as constant) ???? +??? ( terms not containing ?? )???? =??
? ?? 2
(?? -?? )???? +? -4?? 3
???? =0
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
=0
?? (0) =1??? =-1
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
+1 =0 is the final solution.
2.2 Consider the differential equation.
where ?? is a constant. Show that
Page 2
Edurev123
2. Equation of 1st Order and 1st Degree
2.1 Solve :
????
????
=
?? ?? (?? -?? )
?? ?? ?? ?? -?? ?? ?? -?? ?? ?? ,?? (?? )=??
(2009 : 20 Marks)
Solution:
Approach : We check for exactness and find it to be so.
????
????
=
?? 2
(?? -?? )
3?? ?? 2
-?? 2
?? -4?? 3
??? 2
(?? -?? )???? +(3?? ?? 2
-?? 2
?? -4?? 3
)???? =0
Comparing to ?????? +?????? =0
?? =?? 2
(?? -?? )
?? =3?? ?? 2
-?? 2
?? -4?? 3
??? ??? =3?? 2
-2????
??? ??? =3?? 2
-2????
?
??? ??? =
??? ??? , so the equation is exact.
Solution of an exact equation :
??? (treating ?? as constant) ???? +??? ( terms not containing ?? )???? =??
? ?? 2
(?? -?? )???? +? -4?? 3
???? =0
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
=0
?? (0) =1??? =-1
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
+1 =0 is the final solution.
2.2 Consider the differential equation.
where ?? is a constant. Show that
?? '
=???? ,?? >??
(i) if ?? (?? ) is any solution and ?? (?? )=?? (?? )?? -????
, then ?? (?? ) is a constant.
(ii) if ?? <?? , then every solution tends to zero as ?? ?8.
(2010: 12 Marks).
Solution:
(i) Given, the differential equation is :
?? '
=????
?
????
????
=???? ?
????
????
-???? =0
Integrating factor ?? ? -?????? =?? -????
? Solution of equation is ?? ·?? -????
=? 0???? + Constant
? ?? ·?? -????
= Constant
Comparing above equation with
?? (?? ) =?? (?? )·?? -????
It can be concluded that ?? (?? )= Constant
(ii) Let ?? be the constant.
So, the solution is ?? =?? ·?? ????
? if ?? <0 and ?? ?8
?? ????
?0
? ?? ?0
? Every solution tends to zero as ?? ?8 when ?? <0.
2.3 Show that the differential equation
(?? ?? ?? -?? )+?? ?? (?? ?? -?? ?? )?? =??
admits an integrating factor which is a function of (?? +?? ?? ) . Hence, solve the
equation.
(2010 : 12 Marks)
Page 3
Edurev123
2. Equation of 1st Order and 1st Degree
2.1 Solve :
????
????
=
?? ?? (?? -?? )
?? ?? ?? ?? -?? ?? ?? -?? ?? ?? ,?? (?? )=??
(2009 : 20 Marks)
Solution:
Approach : We check for exactness and find it to be so.
????
????
=
?? 2
(?? -?? )
3?? ?? 2
-?? 2
?? -4?? 3
??? 2
(?? -?? )???? +(3?? ?? 2
-?? 2
?? -4?? 3
)???? =0
Comparing to ?????? +?????? =0
?? =?? 2
(?? -?? )
?? =3?? ?? 2
-?? 2
?? -4?? 3
??? ??? =3?? 2
-2????
??? ??? =3?? 2
-2????
?
??? ??? =
??? ??? , so the equation is exact.
Solution of an exact equation :
??? (treating ?? as constant) ???? +??? ( terms not containing ?? )???? =??
? ?? 2
(?? -?? )???? +? -4?? 3
???? =0
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
=0
?? (0) =1??? =-1
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
+1 =0 is the final solution.
2.2 Consider the differential equation.
where ?? is a constant. Show that
?? '
=???? ,?? >??
(i) if ?? (?? ) is any solution and ?? (?? )=?? (?? )?? -????
, then ?? (?? ) is a constant.
(ii) if ?? <?? , then every solution tends to zero as ?? ?8.
(2010: 12 Marks).
Solution:
(i) Given, the differential equation is :
?? '
=????
?
????
????
=???? ?
????
????
-???? =0
Integrating factor ?? ? -?????? =?? -????
? Solution of equation is ?? ·?? -????
=? 0???? + Constant
? ?? ·?? -????
= Constant
Comparing above equation with
?? (?? ) =?? (?? )·?? -????
It can be concluded that ?? (?? )= Constant
(ii) Let ?? be the constant.
So, the solution is ?? =?? ·?? ????
? if ?? <0 and ?? ?8
?? ????
?0
? ?? ?0
? Every solution tends to zero as ?? ?8 when ?? <0.
2.3 Show that the differential equation
(?? ?? ?? -?? )+?? ?? (?? ?? -?? ?? )?? =??
admits an integrating factor which is a function of (?? +?? ?? ) . Hence, solve the
equation.
(2010 : 12 Marks)
Solution:
Given the equation is
(3?? 2
-?? )+2?? (?? 2
-3?? )?? =0
or (3?? 2
-?? )???? +2?? (?? 2
-3?? )???? =0
?????? ,?????? ?? =?? +?? 2
?? 2
=?? -?? and 2?????? =???? -????
? Equation becomes (3(?? -?? )-?? )???? +(?? -?? -3?? )×(???? -???? )=0
? (3?? -4?? )???? +(?? -4?? )(???? -???? )=0
? 3?????? -4?????? +?????? -4?????? -?????? +4?????? =0
? 2?????? +(?? -4?? )???? =0
Now, if integrating factor is a function of ?? +?? 2
, then equation becomes
2???? (?? )???? +(?? -4?? )?? (?? )???? =0
? 2???? (?? )???? +(?? -4?? )?? (?? )???? =
??? ??? ???? +
??? ??? ???? =????
Comparing LHS & RHS, we get
and
??? ??? =2???? (?? )?
?
2
?? ??? ??? =2?? (?? )+2?? ??? ???
and
??? ??? =(?? -4?? )?? (?? )?
??? ??? ??? =-4?? (?? )
? 2?? (?? )+2?? ??? ??? =-4?? (?? )
? 6?? (?? )+2?? ??? ??? =0
? ?? ??? ??? =-3?? ? -3
??? ?? =
??? ??
Integrating both sides, we get
?? (?? )=
?? ?? 3
=
?? (?? +?? 2
)
3
where ?? is a constant.
Page 4
Edurev123
2. Equation of 1st Order and 1st Degree
2.1 Solve :
????
????
=
?? ?? (?? -?? )
?? ?? ?? ?? -?? ?? ?? -?? ?? ?? ,?? (?? )=??
(2009 : 20 Marks)
Solution:
Approach : We check for exactness and find it to be so.
????
????
=
?? 2
(?? -?? )
3?? ?? 2
-?? 2
?? -4?? 3
??? 2
(?? -?? )???? +(3?? ?? 2
-?? 2
?? -4?? 3
)???? =0
Comparing to ?????? +?????? =0
?? =?? 2
(?? -?? )
?? =3?? ?? 2
-?? 2
?? -4?? 3
??? ??? =3?? 2
-2????
??? ??? =3?? 2
-2????
?
??? ??? =
??? ??? , so the equation is exact.
Solution of an exact equation :
??? (treating ?? as constant) ???? +??? ( terms not containing ?? )???? =??
? ?? 2
(?? -?? )???? +? -4?? 3
???? =0
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
=0
?? (0) =1??? =-1
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
+1 =0 is the final solution.
2.2 Consider the differential equation.
where ?? is a constant. Show that
?? '
=???? ,?? >??
(i) if ?? (?? ) is any solution and ?? (?? )=?? (?? )?? -????
, then ?? (?? ) is a constant.
(ii) if ?? <?? , then every solution tends to zero as ?? ?8.
(2010: 12 Marks).
Solution:
(i) Given, the differential equation is :
?? '
=????
?
????
????
=???? ?
????
????
-???? =0
Integrating factor ?? ? -?????? =?? -????
? Solution of equation is ?? ·?? -????
=? 0???? + Constant
? ?? ·?? -????
= Constant
Comparing above equation with
?? (?? ) =?? (?? )·?? -????
It can be concluded that ?? (?? )= Constant
(ii) Let ?? be the constant.
So, the solution is ?? =?? ·?? ????
? if ?? <0 and ?? ?8
?? ????
?0
? ?? ?0
? Every solution tends to zero as ?? ?8 when ?? <0.
2.3 Show that the differential equation
(?? ?? ?? -?? )+?? ?? (?? ?? -?? ?? )?? =??
admits an integrating factor which is a function of (?? +?? ?? ) . Hence, solve the
equation.
(2010 : 12 Marks)
Solution:
Given the equation is
(3?? 2
-?? )+2?? (?? 2
-3?? )?? =0
or (3?? 2
-?? )???? +2?? (?? 2
-3?? )???? =0
?????? ,?????? ?? =?? +?? 2
?? 2
=?? -?? and 2?????? =???? -????
? Equation becomes (3(?? -?? )-?? )???? +(?? -?? -3?? )×(???? -???? )=0
? (3?? -4?? )???? +(?? -4?? )(???? -???? )=0
? 3?????? -4?????? +?????? -4?????? -?????? +4?????? =0
? 2?????? +(?? -4?? )???? =0
Now, if integrating factor is a function of ?? +?? 2
, then equation becomes
2???? (?? )???? +(?? -4?? )?? (?? )???? =0
? 2???? (?? )???? +(?? -4?? )?? (?? )???? =
??? ??? ???? +
??? ??? ???? =????
Comparing LHS & RHS, we get
and
??? ??? =2???? (?? )?
?
2
?? ??? ??? =2?? (?? )+2?? ??? ???
and
??? ??? =(?? -4?? )?? (?? )?
??? ??? ??? =-4?? (?? )
? 2?? (?? )+2?? ??? ??? =-4?? (?? )
? 6?? (?? )+2?? ??? ??? =0
? ?? ??? ??? =-3?? ? -3
??? ?? =
??? ??
Integrating both sides, we get
?? (?? )=
?? ?? 3
=
?? (?? +?? 2
)
3
where ?? is a constant.
? integrating factor is
?? (?? +?? 2
)=
1
(?? +?? 2
)
3
Multiplying eqn, by
1
?? 3
, we get
2?? ?? 3
???? +(?? -4?? )
????
?? 3
=0
? ?? (
2?? -?? ?? 2
) =0
?
2?? -?? ?? 2
=?? , where ?? is a constant.
? Solution of equation is
? 2?? -?? -?? ?? 2
=0
? 2?? -(?? +?? 2
)-?? (?? +?? 2
)
2
=0
? ?? -?? 2
-?? (?? +?? 2
)
2
=0, where ?? is a constant, is the required solution of the
equation.
2.4 Verify that:
?? ?? (???? +???? )?? (??????
?? ???? )+
?? ?? (???? -???? )?? (??????
??
?? ?? )=?????? +??????
Hence show that:
(i) if the differential equation ?????? +?????? =?? is homogenous then (???? +???? )
-?? is
an integrating factor unless ?? ?? +???? =?? .
(ii) if the differential equation ?????? +?????? =?? is not exact but is of the form
?? ?? (?? ,?? )?????? +?? ?? (?? ,?? )?????? =??
then (???? -???? )
-?? is an integrating factor unless ???? -???? =?? .
Solution:
Given:
1
2
(???? +???? )?? (log
?? ?? ?? )+
1
2
(???? -???? )?? (log
??
?? ?? )=?????? +??????
Taking LHS :
Page 5
Edurev123
2. Equation of 1st Order and 1st Degree
2.1 Solve :
????
????
=
?? ?? (?? -?? )
?? ?? ?? ?? -?? ?? ?? -?? ?? ?? ,?? (?? )=??
(2009 : 20 Marks)
Solution:
Approach : We check for exactness and find it to be so.
????
????
=
?? 2
(?? -?? )
3?? ?? 2
-?? 2
?? -4?? 3
??? 2
(?? -?? )???? +(3?? ?? 2
-?? 2
?? -4?? 3
)???? =0
Comparing to ?????? +?????? =0
?? =?? 2
(?? -?? )
?? =3?? ?? 2
-?? 2
?? -4?? 3
??? ??? =3?? 2
-2????
??? ??? =3?? 2
-2????
?
??? ??? =
??? ??? , so the equation is exact.
Solution of an exact equation :
??? (treating ?? as constant) ???? +??? ( terms not containing ?? )???? =??
? ?? 2
(?? -?? )???? +? -4?? 3
???? =0
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
=0
?? (0) =1??? =-1
? ?? 3
?? -
?? 2
?? 2
2
-?? 4
+1 =0 is the final solution.
2.2 Consider the differential equation.
where ?? is a constant. Show that
?? '
=???? ,?? >??
(i) if ?? (?? ) is any solution and ?? (?? )=?? (?? )?? -????
, then ?? (?? ) is a constant.
(ii) if ?? <?? , then every solution tends to zero as ?? ?8.
(2010: 12 Marks).
Solution:
(i) Given, the differential equation is :
?? '
=????
?
????
????
=???? ?
????
????
-???? =0
Integrating factor ?? ? -?????? =?? -????
? Solution of equation is ?? ·?? -????
=? 0???? + Constant
? ?? ·?? -????
= Constant
Comparing above equation with
?? (?? ) =?? (?? )·?? -????
It can be concluded that ?? (?? )= Constant
(ii) Let ?? be the constant.
So, the solution is ?? =?? ·?? ????
? if ?? <0 and ?? ?8
?? ????
?0
? ?? ?0
? Every solution tends to zero as ?? ?8 when ?? <0.
2.3 Show that the differential equation
(?? ?? ?? -?? )+?? ?? (?? ?? -?? ?? )?? =??
admits an integrating factor which is a function of (?? +?? ?? ) . Hence, solve the
equation.
(2010 : 12 Marks)
Solution:
Given the equation is
(3?? 2
-?? )+2?? (?? 2
-3?? )?? =0
or (3?? 2
-?? )???? +2?? (?? 2
-3?? )???? =0
?????? ,?????? ?? =?? +?? 2
?? 2
=?? -?? and 2?????? =???? -????
? Equation becomes (3(?? -?? )-?? )???? +(?? -?? -3?? )×(???? -???? )=0
? (3?? -4?? )???? +(?? -4?? )(???? -???? )=0
? 3?????? -4?????? +?????? -4?????? -?????? +4?????? =0
? 2?????? +(?? -4?? )???? =0
Now, if integrating factor is a function of ?? +?? 2
, then equation becomes
2???? (?? )???? +(?? -4?? )?? (?? )???? =0
? 2???? (?? )???? +(?? -4?? )?? (?? )???? =
??? ??? ???? +
??? ??? ???? =????
Comparing LHS & RHS, we get
and
??? ??? =2???? (?? )?
?
2
?? ??? ??? =2?? (?? )+2?? ??? ???
and
??? ??? =(?? -4?? )?? (?? )?
??? ??? ??? =-4?? (?? )
? 2?? (?? )+2?? ??? ??? =-4?? (?? )
? 6?? (?? )+2?? ??? ??? =0
? ?? ??? ??? =-3?? ? -3
??? ?? =
??? ??
Integrating both sides, we get
?? (?? )=
?? ?? 3
=
?? (?? +?? 2
)
3
where ?? is a constant.
? integrating factor is
?? (?? +?? 2
)=
1
(?? +?? 2
)
3
Multiplying eqn, by
1
?? 3
, we get
2?? ?? 3
???? +(?? -4?? )
????
?? 3
=0
? ?? (
2?? -?? ?? 2
) =0
?
2?? -?? ?? 2
=?? , where ?? is a constant.
? Solution of equation is
? 2?? -?? -?? ?? 2
=0
? 2?? -(?? +?? 2
)-?? (?? +?? 2
)
2
=0
? ?? -?? 2
-?? (?? +?? 2
)
2
=0, where ?? is a constant, is the required solution of the
equation.
2.4 Verify that:
?? ?? (???? +???? )?? (??????
?? ???? )+
?? ?? (???? -???? )?? (??????
??
?? ?? )=?????? +??????
Hence show that:
(i) if the differential equation ?????? +?????? =?? is homogenous then (???? +???? )
-?? is
an integrating factor unless ?? ?? +???? =?? .
(ii) if the differential equation ?????? +?????? =?? is not exact but is of the form
?? ?? (?? ,?? )?????? +?? ?? (?? ,?? )?????? =??
then (???? -???? )
-?? is an integrating factor unless ???? -???? =?? .
Solution:
Given:
1
2
(???? +???? )?? (log
?? ?? ?? )+
1
2
(???? -???? )?? (log
??
?? ?? )=?????? +??????
Taking LHS :
1
2
(???? +???? )×
1
????
(?????? +?????? )+
1
2
(???? -???? )×
?? ?? ×
(?????? -?????? )
?? 2
=
1
2
[(???? +???? )(
????
?? +
????
?? )+(???? -???? )(
????
?? -
????
?? )] (1)
=
1
2
????
?? ?? ?? +
1
2
????
????
?? +
????
2
????
?? +
????
2
????
?? +
1
2
????
????
?? +
1
2
????
????
?? -
????
2
????
?? -
????
2
????
?? =?????? + Ndy = RHS
Hence, given equation is verified.
(i) Lot ?????? +?????? =0 be a homogeneous equation.
? ?????? +?????? =
1
2
[(???? +???? )(
????
?? +
????
?? )+(???? -???? )(
????
?? -
????
?? )] ???????? (1)
If
1
???? +????
is an integrating factor, then equation becomes
?????? +??????
???? +????
=
1
2
[
?????? +?????? ????
+
???? -????
???? +????
×
(?????? -?????? )
????
]
?
?????? +??????
???? +????
=
1
2
(?? log
?? (???? )+?? (
?? ?? )?? log
?? (
?? ?? ))
where ?? (
?? ?? )=
?? -?? (
?? ?? )
?? +?? (
?? ?? )
?
?? ???? +??????
???? +????
=
1
2
(?? (log
?? ???? )+?? (?? log
?? (
?? ?? )
)?? (log
??
?? ?? ))
?
?????? +??????
???? +????
=
1
2
(?? (log
?? ???? )+?? (log
??
?? ?? )?? (log
?? ?? ))
which is an exact differential.
?
1
???? +????
is an integraing factor un less ???? +?????? ?0.
(ii) Rewriting ?????? +?????? as
?????? +?????? =
1
2
[(???? +???? )(
????
?? +
????
?? )+(???? -???? )(
????
?? -
????
?? )] ???????? (1)
if (???? -???? )
-1
is an integrating factor, then equation heromes
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FAQs on Equation of 1st Order and 1st Degree - Mathematics Optional Notes for UPSC
| 1. What is the general form of a first-order and first-degree equation? |  |
Ans. The general form of a first-order and first-degree equation is given by: ax + by = c, where a, b, and c are constants and a and b are not both zero.
| 2. How do you determine if an equation is first order and first degree? | |
Ans. An equation is considered first order and first degree if the highest power of the variable in the equation is 1 and the equation does not contain any higher powers of the variable.
| 3. What are some real-life applications of first-order and first-degree equations? | |
Ans. First-order and first-degree equations are commonly used in various fields such as physics, chemistry, engineering, and economics to describe relationships between variables that are linear in nature.
| 4. How do you solve a first-order and first-degree equation? | |
Ans. To solve a first-order and first-degree equation, you can use methods such as substitution, elimination, or graphing to find the values of the variables that satisfy the equation.
| 5. Can a first-order and first-degree equation have multiple solutions? | |
Ans. Yes, a first-order and first-degree equation can have multiple solutions, depending on the values of the constants in the equation and the nature of the relationship between the variables.
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Nov 23, 2024 Last updated |
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The "Equation of 1st Order and 1st Degree UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
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their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Equation of 1st Order and 1st Degree on the App
Students of UPSC can study Equation of 1st Order and 1st Degree alongwith tests & analysis from the EduRev app,
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The content of Equation of 1st Order and 1st Degree is prepared as per the latest UPSC syllabus.
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