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 Page 1


 Solved Examples on Differential Equation 
JEE Main 
Single Correct Type 
Q1. The order and degree of the differential equation ?? = ?? ????
????
+
v
?? ?? (
????
????
)
?? + ?? ?? are 
(a) 1,2 
(b) 2,1 
(c) 1,1 
(d) 2,2 
Ans: (a) Clearly, highest order derivative involved is 
????
????
, having order 1 . 
Expressing the above differential equation as a polynomial in derivative, we have 
( ?? - ?? ????
????
)
2
= ?? 2
(
????
????
)
2
+ ?? 2
 
i.e., ( ?? 2
- ?? 2
) (
????
????
)
2
- 2 ????
????
????
+ ?? 2
- ?? 2
= 0 
In this equation, the power of highest order derivative is 2 . So its degree is 2 . 
Q2. The degree of the differential equation 
?? ?? ?? ?? ?? ?? + ?? (
????
????
)
?? = ?? ?? ?????? ? (
?? ?? ?? ?? ?? ?? ) is 
(a) 1 
(b) 2 
(c) 3 
(d) None of these 
Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term 
?? 2
log ? (
?? 2
?? ?? ?? 2
). Hence degree of the differential equation is 'not defined'. 
 
 
 
 
 
 
Page 2


 Solved Examples on Differential Equation 
JEE Main 
Single Correct Type 
Q1. The order and degree of the differential equation ?? = ?? ????
????
+
v
?? ?? (
????
????
)
?? + ?? ?? are 
(a) 1,2 
(b) 2,1 
(c) 1,1 
(d) 2,2 
Ans: (a) Clearly, highest order derivative involved is 
????
????
, having order 1 . 
Expressing the above differential equation as a polynomial in derivative, we have 
( ?? - ?? ????
????
)
2
= ?? 2
(
????
????
)
2
+ ?? 2
 
i.e., ( ?? 2
- ?? 2
) (
????
????
)
2
- 2 ????
????
????
+ ?? 2
- ?? 2
= 0 
In this equation, the power of highest order derivative is 2 . So its degree is 2 . 
Q2. The degree of the differential equation 
?? ?? ?? ?? ?? ?? + ?? (
????
????
)
?? = ?? ?? ?????? ? (
?? ?? ?? ?? ?? ?? ) is 
(a) 1 
(b) 2 
(c) 3 
(d) None of these 
Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term 
?? 2
log ? (
?? 2
?? ?? ?? 2
). Hence degree of the differential equation is 'not defined'. 
 
 
 
 
 
 
Q3. Differential equation whose general solution is ?? = ?? ?? ?? +
?? ?? ?? for all values of ?? ?? and ?? ?? is 
(a) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? +
????
????
= ?? 
(b) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? -
????
????
= ?? 
(c) 
?? ?? ?? ?? ?? ?? -
?? ?? ?? ????
????
= ?? 
(d) 
?? ?? ?? ?? ?? ?? +
?? ?? ????
????
-
?? ?? ?? = ?? 
Ans: (d) ?? = ?? 1
?? +
?? 2
?? (i) 
There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. 
Differentiating (i) with respect to ?? , 
????
????
= ?? 1
-
?? 2
?? 2
 
Again differentiating with respect to ?? , 
?? 2
?? ?? ?? 2
=
2 ?? 2
?? 3
 
From (iii), ?? 2
=
?? 3
2
?? 2
?? ?? ?? 2
 and from (ii), ?? 1
=
????
????
+
?? 2
?? 2
; 
? ?? 1
=
????
????
+
?? 2
?? 2
?? ?? ?? 2
 
From (i), ?? = (
????
????
+
?? 2
·
?? 2
?? ?? ?? 2
) ?? +
?? 2
2
·
?? 2
?? ?? ?? 2
? ?? = ?? 2
?? 2
?? ?? ?? 2
+ ?? ????
????
 
?
?? 2
?? ?? ?? 2
+
1
?? ????
????
-
?? ?? 2
= 0 
Q4.  ?? =
?? ?? + ?? is a solution of the differential equation 
(a) ?? ?? ????
????
= ?? ?? 
(b) ?? ?? ????
????
= ?? ?? 
(c) ?? ????
????
= ?? 
(d) ?? ????
????
= ?? 
Ans: (b) We have ?? =
?? ?? + 1
?
1
?? =
?? + 1
?? = 1 +
1
?? 
 Differentiating w.r.t. 
? -
1
?? 2
????
????
= 0 -
1
?? 2
? ? ?? 2
????
????
= ?? 2
 
 
 
Page 3


 Solved Examples on Differential Equation 
JEE Main 
Single Correct Type 
Q1. The order and degree of the differential equation ?? = ?? ????
????
+
v
?? ?? (
????
????
)
?? + ?? ?? are 
(a) 1,2 
(b) 2,1 
(c) 1,1 
(d) 2,2 
Ans: (a) Clearly, highest order derivative involved is 
????
????
, having order 1 . 
Expressing the above differential equation as a polynomial in derivative, we have 
( ?? - ?? ????
????
)
2
= ?? 2
(
????
????
)
2
+ ?? 2
 
i.e., ( ?? 2
- ?? 2
) (
????
????
)
2
- 2 ????
????
????
+ ?? 2
- ?? 2
= 0 
In this equation, the power of highest order derivative is 2 . So its degree is 2 . 
Q2. The degree of the differential equation 
?? ?? ?? ?? ?? ?? + ?? (
????
????
)
?? = ?? ?? ?????? ? (
?? ?? ?? ?? ?? ?? ) is 
(a) 1 
(b) 2 
(c) 3 
(d) None of these 
Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term 
?? 2
log ? (
?? 2
?? ?? ?? 2
). Hence degree of the differential equation is 'not defined'. 
 
 
 
 
 
 
Q3. Differential equation whose general solution is ?? = ?? ?? ?? +
?? ?? ?? for all values of ?? ?? and ?? ?? is 
(a) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? +
????
????
= ?? 
(b) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? -
????
????
= ?? 
(c) 
?? ?? ?? ?? ?? ?? -
?? ?? ?? ????
????
= ?? 
(d) 
?? ?? ?? ?? ?? ?? +
?? ?? ????
????
-
?? ?? ?? = ?? 
Ans: (d) ?? = ?? 1
?? +
?? 2
?? (i) 
There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. 
Differentiating (i) with respect to ?? , 
????
????
= ?? 1
-
?? 2
?? 2
 
Again differentiating with respect to ?? , 
?? 2
?? ?? ?? 2
=
2 ?? 2
?? 3
 
From (iii), ?? 2
=
?? 3
2
?? 2
?? ?? ?? 2
 and from (ii), ?? 1
=
????
????
+
?? 2
?? 2
; 
? ?? 1
=
????
????
+
?? 2
?? 2
?? ?? ?? 2
 
From (i), ?? = (
????
????
+
?? 2
·
?? 2
?? ?? ?? 2
) ?? +
?? 2
2
·
?? 2
?? ?? ?? 2
? ?? = ?? 2
?? 2
?? ?? ?? 2
+ ?? ????
????
 
?
?? 2
?? ?? ?? 2
+
1
?? ????
????
-
?? ?? 2
= 0 
Q4.  ?? =
?? ?? + ?? is a solution of the differential equation 
(a) ?? ?? ????
????
= ?? ?? 
(b) ?? ?? ????
????
= ?? ?? 
(c) ?? ????
????
= ?? 
(d) ?? ????
????
= ?? 
Ans: (b) We have ?? =
?? ?? + 1
?
1
?? =
?? + 1
?? = 1 +
1
?? 
 Differentiating w.r.t. 
? -
1
?? 2
????
????
= 0 -
1
?? 2
? ? ?? 2
????
????
= ?? 2
 
 
 
Q5. The differential equation of family of curves whose tangent form an angle of ?? / ?? with the 
hyperbola ???? = ?? ?? is 
(a) 
????
????
=
?? ?? + ?? ?? ?? ?? - ?? ?? 
(b) 
????
????
=
?? ?? - ?? ?? ?? ?? + ?? ?? 
(c) 
????
????
= -
?? ?? ?? ?? 
(d) None of these 
Ans: (b) The slope of the tangent to the family of curves is ?? 1
=
????
????
 
Equation of the hyperbola is ???? = ?? 2
? ?? =
?? 2
?? 
?
????
????
= -
?? 2
?? 2
 
? Slope of tangent to ???? = ?? 2
 is ?? 2
= -
?? 2
?? 2
 
Now tan ?
?? 4
=
?? 1
- ?? 2
1 + ?? 1
?? 2
? 1 =
????
????
+
?? 2
?? 2
1 -
?? 2
?? 2
????
????
?
????
????
( 1 +
?? 2
?? 2
) = ( 1 -
?? 2
?? 2
) 
?
????
????
=
?? 2
- ?? 2
?? 2
+ ?? 2
 
 
Q5.  The solution of the differential equation ( ?? + ?? ?? )
????
????
= ?? ( ?? + ?? ?? ) is 
(a) ?? ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? 
(b) ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? 
(c) ?? ???? ?? - ?? ? ?? + ?????? ? ( ?? + ?? ?? ) + ?? = ?? 
(d) None of these 
Ans: (a) Separating the variables, we can re-write the given differential equation as 
???? ?? 1 + ?? 2
=
????
1 + ?? 2
? ? ?
2 ???? ?? 1 + ?? 2
= 2 ? ?
????
1 + ?? 2
? 2 tan
- 1
? ?? = log
?? ? ( 1 + ?? 2
) + ?? 
Q6. The solution of 
????
????
=
?? ?? ?? + ?? ???? ? ?? is 
(a) ?? =
?? ?? ?? - ?? ???? ? ?? + ?? 
(b) ?? + ?? ???? ? ?? = ?? + ?? 
(c) ?? =
?? ?? ?? + ?? ???? ? ?? + ?? 
(d) None of these 
Ans: (a) Given equation may be re -written as ???? = ( ?? 2
+ sin ? ?? ) ???? 
 Integrating, ? ? ???? = ? ? ( ?? 2
+ sin ? ?? ) ????
? ? ?? =
?? 3
3
- c os ? ?? + ?? 
Page 4


 Solved Examples on Differential Equation 
JEE Main 
Single Correct Type 
Q1. The order and degree of the differential equation ?? = ?? ????
????
+
v
?? ?? (
????
????
)
?? + ?? ?? are 
(a) 1,2 
(b) 2,1 
(c) 1,1 
(d) 2,2 
Ans: (a) Clearly, highest order derivative involved is 
????
????
, having order 1 . 
Expressing the above differential equation as a polynomial in derivative, we have 
( ?? - ?? ????
????
)
2
= ?? 2
(
????
????
)
2
+ ?? 2
 
i.e., ( ?? 2
- ?? 2
) (
????
????
)
2
- 2 ????
????
????
+ ?? 2
- ?? 2
= 0 
In this equation, the power of highest order derivative is 2 . So its degree is 2 . 
Q2. The degree of the differential equation 
?? ?? ?? ?? ?? ?? + ?? (
????
????
)
?? = ?? ?? ?????? ? (
?? ?? ?? ?? ?? ?? ) is 
(a) 1 
(b) 2 
(c) 3 
(d) None of these 
Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term 
?? 2
log ? (
?? 2
?? ?? ?? 2
). Hence degree of the differential equation is 'not defined'. 
 
 
 
 
 
 
Q3. Differential equation whose general solution is ?? = ?? ?? ?? +
?? ?? ?? for all values of ?? ?? and ?? ?? is 
(a) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? +
????
????
= ?? 
(b) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? -
????
????
= ?? 
(c) 
?? ?? ?? ?? ?? ?? -
?? ?? ?? ????
????
= ?? 
(d) 
?? ?? ?? ?? ?? ?? +
?? ?? ????
????
-
?? ?? ?? = ?? 
Ans: (d) ?? = ?? 1
?? +
?? 2
?? (i) 
There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. 
Differentiating (i) with respect to ?? , 
????
????
= ?? 1
-
?? 2
?? 2
 
Again differentiating with respect to ?? , 
?? 2
?? ?? ?? 2
=
2 ?? 2
?? 3
 
From (iii), ?? 2
=
?? 3
2
?? 2
?? ?? ?? 2
 and from (ii), ?? 1
=
????
????
+
?? 2
?? 2
; 
? ?? 1
=
????
????
+
?? 2
?? 2
?? ?? ?? 2
 
From (i), ?? = (
????
????
+
?? 2
·
?? 2
?? ?? ?? 2
) ?? +
?? 2
2
·
?? 2
?? ?? ?? 2
? ?? = ?? 2
?? 2
?? ?? ?? 2
+ ?? ????
????
 
?
?? 2
?? ?? ?? 2
+
1
?? ????
????
-
?? ?? 2
= 0 
Q4.  ?? =
?? ?? + ?? is a solution of the differential equation 
(a) ?? ?? ????
????
= ?? ?? 
(b) ?? ?? ????
????
= ?? ?? 
(c) ?? ????
????
= ?? 
(d) ?? ????
????
= ?? 
Ans: (b) We have ?? =
?? ?? + 1
?
1
?? =
?? + 1
?? = 1 +
1
?? 
 Differentiating w.r.t. 
? -
1
?? 2
????
????
= 0 -
1
?? 2
? ? ?? 2
????
????
= ?? 2
 
 
 
Q5. The differential equation of family of curves whose tangent form an angle of ?? / ?? with the 
hyperbola ???? = ?? ?? is 
(a) 
????
????
=
?? ?? + ?? ?? ?? ?? - ?? ?? 
(b) 
????
????
=
?? ?? - ?? ?? ?? ?? + ?? ?? 
(c) 
????
????
= -
?? ?? ?? ?? 
(d) None of these 
Ans: (b) The slope of the tangent to the family of curves is ?? 1
=
????
????
 
Equation of the hyperbola is ???? = ?? 2
? ?? =
?? 2
?? 
?
????
????
= -
?? 2
?? 2
 
? Slope of tangent to ???? = ?? 2
 is ?? 2
= -
?? 2
?? 2
 
Now tan ?
?? 4
=
?? 1
- ?? 2
1 + ?? 1
?? 2
? 1 =
????
????
+
?? 2
?? 2
1 -
?? 2
?? 2
????
????
?
????
????
( 1 +
?? 2
?? 2
) = ( 1 -
?? 2
?? 2
) 
?
????
????
=
?? 2
- ?? 2
?? 2
+ ?? 2
 
 
Q5.  The solution of the differential equation ( ?? + ?? ?? )
????
????
= ?? ( ?? + ?? ?? ) is 
(a) ?? ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? 
(b) ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? 
(c) ?? ???? ?? - ?? ? ?? + ?????? ? ( ?? + ?? ?? ) + ?? = ?? 
(d) None of these 
Ans: (a) Separating the variables, we can re-write the given differential equation as 
???? ?? 1 + ?? 2
=
????
1 + ?? 2
? ? ?
2 ???? ?? 1 + ?? 2
= 2 ? ?
????
1 + ?? 2
? 2 tan
- 1
? ?? = log
?? ? ( 1 + ?? 2
) + ?? 
Q6. The solution of 
????
????
=
?? ?? ?? + ?? ???? ? ?? is 
(a) ?? =
?? ?? ?? - ?? ???? ? ?? + ?? 
(b) ?? + ?? ???? ? ?? = ?? + ?? 
(c) ?? =
?? ?? ?? + ?? ???? ? ?? + ?? 
(d) None of these 
Ans: (a) Given equation may be re -written as ???? = ( ?? 2
+ sin ? ?? ) ???? 
 Integrating, ? ? ???? = ? ? ( ?? 2
+ sin ? ?? ) ????
? ? ?? =
?? 3
3
- c os ? ?? + ?? 
Q7.  The solution of the differential equation ?? ????
????
= ?? ( ?????? ? ?? - ?????? ? ?? + ?? ) is 
(a) ?? = ?? ?? ????
 
(b) ?? + ?? ?? ????
= ?? 
(c) ?? + ?? ?? = ?? 
(d) None of these 
Ans: (a) Given equation may be expressed as 
????
????
=
?? ?? [ log ? (
?? ?? ) + 1 ] 
 Let 
?? ?? = ?? ? ?? = ???? ?
????
????
= ?? + ?? ????
????
? ? From (i), ?? + ?? ????
????
= ?? ( log ? ?? + 1 ) ? ?? ????
????
= ?? log ? ?? ?
????
?? log ? ?? =
????
?? ? ? ?
1
log ? ?? ?? ( log ? ?? ) = ? ?
????
?? ? ? log ? ( log ? ?? ) = log ? ?? + log ? ?? ? log ? ( log ? ?? ) = log ? ( ???? ) ? log ? ?? = ???? ? ?? = ?? ????
?
?? ?? = ?? ????
, ? ?? = ?? ?? ????
 
 
Q8.  The general solution of the differential equation ( ?? + ?? ) ???? + ???? ?? = ?? is 
(a) ?? ?? + ?? ?? = ?? 
(b) ?? ?? ?? - ?? ?? = ?? 
(c) ?? ?? + ?? ???? = ?? 
(d) ?? ?? + ?? ???? = ?? 
Ans: (c) We have ???? ?? + ( ?? ?? ?? + ???? ?? ) = 0 ? ???? ?? + ?? ( ???? ) = 0 
Integrating, 
?? 2
2
+ ???? =
?? 2
 
? ?? 2
+ 2 ???? = ?? 
 
Q9.  Which of the following is a linear differential equation 
(a) (
?? ?? ?? ?? ?? ?? )
?? + ?? ?? (
????
????
)
?? = ?? 
(b) ?? =
????
????
+
v
?? + (
????
????
)
?? 
(c) 
????
????
+
?? ?? = ?????? ? ?? 
(d) ?? ????
????
- ?? = ?? 
Ans: (c) (a), (b), (d) do not fulfill the criteria of a linear differential equation but (c) do. 
????
????
+
?? ?? =
log ? ?? is a linear differential equation. 
 
 
 
Q10.  Find the integral factor of equation ( ?? ?? + ?? )
????
????
+ ?? ???? = ?? ?? - ?? 
 
Page 5


 Solved Examples on Differential Equation 
JEE Main 
Single Correct Type 
Q1. The order and degree of the differential equation ?? = ?? ????
????
+
v
?? ?? (
????
????
)
?? + ?? ?? are 
(a) 1,2 
(b) 2,1 
(c) 1,1 
(d) 2,2 
Ans: (a) Clearly, highest order derivative involved is 
????
????
, having order 1 . 
Expressing the above differential equation as a polynomial in derivative, we have 
( ?? - ?? ????
????
)
2
= ?? 2
(
????
????
)
2
+ ?? 2
 
i.e., ( ?? 2
- ?? 2
) (
????
????
)
2
- 2 ????
????
????
+ ?? 2
- ?? 2
= 0 
In this equation, the power of highest order derivative is 2 . So its degree is 2 . 
Q2. The degree of the differential equation 
?? ?? ?? ?? ?? ?? + ?? (
????
????
)
?? = ?? ?? ?????? ? (
?? ?? ?? ?? ?? ?? ) is 
(a) 1 
(b) 2 
(c) 3 
(d) None of these 
Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term 
?? 2
log ? (
?? 2
?? ?? ?? 2
). Hence degree of the differential equation is 'not defined'. 
 
 
 
 
 
 
Q3. Differential equation whose general solution is ?? = ?? ?? ?? +
?? ?? ?? for all values of ?? ?? and ?? ?? is 
(a) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? +
????
????
= ?? 
(b) 
?? ?? ?? ?? ?? ?? +
?? ?? ?? -
????
????
= ?? 
(c) 
?? ?? ?? ?? ?? ?? -
?? ?? ?? ????
????
= ?? 
(d) 
?? ?? ?? ?? ?? ?? +
?? ?? ????
????
-
?? ?? ?? = ?? 
Ans: (d) ?? = ?? 1
?? +
?? 2
?? (i) 
There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. 
Differentiating (i) with respect to ?? , 
????
????
= ?? 1
-
?? 2
?? 2
 
Again differentiating with respect to ?? , 
?? 2
?? ?? ?? 2
=
2 ?? 2
?? 3
 
From (iii), ?? 2
=
?? 3
2
?? 2
?? ?? ?? 2
 and from (ii), ?? 1
=
????
????
+
?? 2
?? 2
; 
? ?? 1
=
????
????
+
?? 2
?? 2
?? ?? ?? 2
 
From (i), ?? = (
????
????
+
?? 2
·
?? 2
?? ?? ?? 2
) ?? +
?? 2
2
·
?? 2
?? ?? ?? 2
? ?? = ?? 2
?? 2
?? ?? ?? 2
+ ?? ????
????
 
?
?? 2
?? ?? ?? 2
+
1
?? ????
????
-
?? ?? 2
= 0 
Q4.  ?? =
?? ?? + ?? is a solution of the differential equation 
(a) ?? ?? ????
????
= ?? ?? 
(b) ?? ?? ????
????
= ?? ?? 
(c) ?? ????
????
= ?? 
(d) ?? ????
????
= ?? 
Ans: (b) We have ?? =
?? ?? + 1
?
1
?? =
?? + 1
?? = 1 +
1
?? 
 Differentiating w.r.t. 
? -
1
?? 2
????
????
= 0 -
1
?? 2
? ? ?? 2
????
????
= ?? 2
 
 
 
Q5. The differential equation of family of curves whose tangent form an angle of ?? / ?? with the 
hyperbola ???? = ?? ?? is 
(a) 
????
????
=
?? ?? + ?? ?? ?? ?? - ?? ?? 
(b) 
????
????
=
?? ?? - ?? ?? ?? ?? + ?? ?? 
(c) 
????
????
= -
?? ?? ?? ?? 
(d) None of these 
Ans: (b) The slope of the tangent to the family of curves is ?? 1
=
????
????
 
Equation of the hyperbola is ???? = ?? 2
? ?? =
?? 2
?? 
?
????
????
= -
?? 2
?? 2
 
? Slope of tangent to ???? = ?? 2
 is ?? 2
= -
?? 2
?? 2
 
Now tan ?
?? 4
=
?? 1
- ?? 2
1 + ?? 1
?? 2
? 1 =
????
????
+
?? 2
?? 2
1 -
?? 2
?? 2
????
????
?
????
????
( 1 +
?? 2
?? 2
) = ( 1 -
?? 2
?? 2
) 
?
????
????
=
?? 2
- ?? 2
?? 2
+ ?? 2
 
 
Q5.  The solution of the differential equation ( ?? + ?? ?? )
????
????
= ?? ( ?? + ?? ?? ) is 
(a) ?? ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? 
(b) ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? 
(c) ?? ???? ?? - ?? ? ?? + ?????? ? ( ?? + ?? ?? ) + ?? = ?? 
(d) None of these 
Ans: (a) Separating the variables, we can re-write the given differential equation as 
???? ?? 1 + ?? 2
=
????
1 + ?? 2
? ? ?
2 ???? ?? 1 + ?? 2
= 2 ? ?
????
1 + ?? 2
? 2 tan
- 1
? ?? = log
?? ? ( 1 + ?? 2
) + ?? 
Q6. The solution of 
????
????
=
?? ?? ?? + ?? ???? ? ?? is 
(a) ?? =
?? ?? ?? - ?? ???? ? ?? + ?? 
(b) ?? + ?? ???? ? ?? = ?? + ?? 
(c) ?? =
?? ?? ?? + ?? ???? ? ?? + ?? 
(d) None of these 
Ans: (a) Given equation may be re -written as ???? = ( ?? 2
+ sin ? ?? ) ???? 
 Integrating, ? ? ???? = ? ? ( ?? 2
+ sin ? ?? ) ????
? ? ?? =
?? 3
3
- c os ? ?? + ?? 
Q7.  The solution of the differential equation ?? ????
????
= ?? ( ?????? ? ?? - ?????? ? ?? + ?? ) is 
(a) ?? = ?? ?? ????
 
(b) ?? + ?? ?? ????
= ?? 
(c) ?? + ?? ?? = ?? 
(d) None of these 
Ans: (a) Given equation may be expressed as 
????
????
=
?? ?? [ log ? (
?? ?? ) + 1 ] 
 Let 
?? ?? = ?? ? ?? = ???? ?
????
????
= ?? + ?? ????
????
? ? From (i), ?? + ?? ????
????
= ?? ( log ? ?? + 1 ) ? ?? ????
????
= ?? log ? ?? ?
????
?? log ? ?? =
????
?? ? ? ?
1
log ? ?? ?? ( log ? ?? ) = ? ?
????
?? ? ? log ? ( log ? ?? ) = log ? ?? + log ? ?? ? log ? ( log ? ?? ) = log ? ( ???? ) ? log ? ?? = ???? ? ?? = ?? ????
?
?? ?? = ?? ????
, ? ?? = ?? ?? ????
 
 
Q8.  The general solution of the differential equation ( ?? + ?? ) ???? + ???? ?? = ?? is 
(a) ?? ?? + ?? ?? = ?? 
(b) ?? ?? ?? - ?? ?? = ?? 
(c) ?? ?? + ?? ???? = ?? 
(d) ?? ?? + ?? ???? = ?? 
Ans: (c) We have ???? ?? + ( ?? ?? ?? + ???? ?? ) = 0 ? ???? ?? + ?? ( ???? ) = 0 
Integrating, 
?? 2
2
+ ???? =
?? 2
 
? ?? 2
+ 2 ???? = ?? 
 
Q9.  Which of the following is a linear differential equation 
(a) (
?? ?? ?? ?? ?? ?? )
?? + ?? ?? (
????
????
)
?? = ?? 
(b) ?? =
????
????
+
v
?? + (
????
????
)
?? 
(c) 
????
????
+
?? ?? = ?????? ? ?? 
(d) ?? ????
????
- ?? = ?? 
Ans: (c) (a), (b), (d) do not fulfill the criteria of a linear differential equation but (c) do. 
????
????
+
?? ?? =
log ? ?? is a linear differential equation. 
 
 
 
Q10.  Find the integral factor of equation ( ?? ?? + ?? )
????
????
+ ?? ???? = ?? ?? - ?? 
 
(a) ?? ?? + ?? 
(b) 
?? ?? ?? ?? + ?? 
(c) 
?? ?? - ?? ?? ?? + ?? 
(d) None of these 
Ans: (a) Given equation may be written as 
????
????
+
2 ?? ?? 2
+ 1
?? =
?? 2
- 1
?? 2
+ 1
 
Comparing with 
????
????
+ ???? = ?? , 
?? =
2 ?? ?? 2
+ 1
 I.F. = ?? ? ? ?? ???? = ?? ? ?
2 ?? ???? 1 + ?? 2
= ?? ln ? ( 1 + ?? 2
)
= 1 + ?? 2
 
 
Q11.  The solution of the differential equation ( ?? + ?? ?? ) + ( ?? - ?? ?? ???? - ?? ? ?? )
????
????
= ?? is 
(a) ( ?? - ?? ) = ?? ?? ?? ???? - ?? ? ?? 
(b) ?? ?? ?? ?? ???? - ?? ? ?? = ?? ?? ?? ???? - ?? ? ?? + ?? 
(c) ?? ?? ?? ???? - ?? ? ?? = ???? ?? - ?? ? ?? + ?? 
(d) ?? ?? ?? ?? ???? - ?? ? ?? = ?? ?? ???? - ?? ? ?? + ?? 
Ans: (b) We have ( ?? - ?? tan
- 1
? ?? )
????
????
= - ( 1 + ?? 2
) ?
????
????
= - (
?? - ?? t a n
- 1
? ?? 1 + ?? 2
) ?
????
????
+
1
1 + ?? 2
?? =
?? t a n
- 1
? ?? 1 + ?? 2
 
This is a linear differential equation of the form 
????
????
+ ?? ( ?? ) · ?? = ?? ( ?? ) 
?? =
1
1 + ?? 2
, ? ?? =
?? tan
- 1
? ?? 1 + ?? 2
 Integrating factor = ?? ? ? ?????? = ?? ? ?
????
1 + ?? 2
= ?? tan
- 1
? ?? 
Multiplying (i) by I.F. and integrating, ?? ?? tan
- 1
? ?? = ?
?? t a n
- 1
? ?? 1 + ?? 2
· ?? tan
- 1
? ?? ???? = ?
( ?? t a n
- 1
? ?? )
2
????
1 + ?? 2
=
( ?? t a n
- 1
? ?? )
2
2
+
?? 2
 
? 2 ?? ?? tan
- 1
? ?? = ?? 2 tan
- 1
? ?? + ?? 
 
 
Q12.  Solution of 
????
????
- ?? ???? ?? ? ?? = - ?? ?? ?? ???? ? ?? is 
(a) ?? ?? ???? ? ?? = ???? ?? ? ?? + ?? 
(b) 
?? ?? ?? ? ?? ?? = ???? ?? ? ?? + ?? 
(c) ?? ?? ???? ? ?? = ???? ?? ? ?? + ?? ? (d) None of these

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