Differential Equation (Solved MCQs and Notes)

Notes 36 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

1
Unit - 10
Differential Equation
Important Points
Differential Equation :
“y = f(x) and the derivatives of w.r.t. x are
2 3
2 3
, ,
dy d y d y
dx
dx dx
,........... then the functional
equation F(x, y,
2
2
,
dy d y
dx
dx
.......) = 0 is called an ordinary differential equation.”
Example, (1)
3 2
2
3 2
d y dy d y
x y log
dx dx dx
 
 
 
 
 
 
 
(2)
2
2
 
dy d y
og xy
dx
dx
Order of a differential equation :
“Order of the highest order derivative of the dependent variable with respect to the
independent variable occurring in a given differential equation is called the order of
differential equation.”
Example, (1) order of
2
5
3
3
 
 
  
 
 
 
 
 
d y dy
x y o
dx
dx
is 3 -
(2) order of
2
2
dy
dx
d y
e
dx

is 2 -
Degree of a differential equation :
“When a differential equation is in a polynomial form in derivatives, the highest power of
the highest order derivative occurring in the differential equation is called the degree of
the differential equation.”
Note : (1) The degree of a differential equation is a positive integer.
(2) If the differential equation cannot be expressed in a polynomial form in the deriva-
tives, the degree of the differential equation is not defined.
Example : (1) The degree of
3
2
2
dy d y
y
dx
dx
 
 
 
 
is 1 -
(2) The degree of
2
2
sin
d y dy
x
dx
dx

=o is not defined.
Page 1
2
Differential Equation of first order and first degree :
f(x, y) dx + g(x, y) dy = 0 OR
dy
dx
=F(x.y) is form of first order and first degree differential
equation.
(1) Differential Equation of variables separable :

p(x).dx + q(y).dy = 0 equation is said to be in variables separable form.

solution : p(x).dx + q(y).dy= 0
( ) ( )  
 
p x dx q y dy c
is the general solution (c is an arbitrang constant)
(2) Homogeneous differential equation :

If in a differential equation f(x, y) dx + g(x, y) dy = 0, f(x, y) and g(x, y) are homo-
geneous functions with same degree, then this defferential equation is called ho-
mogeneous differential equation.
The homogenous differential equation be in the form of
 

 
 

dy y
dx x

Solution : Let
y
x
 
 

y x
  

dy dy
x
dx dx

Differential equation,
 
  

  
d
x
dx
 
 


  
d dx
x
(variable separable form)
 
1 1
 

 

  
d dx
x
 
1
  




  
d og x c
This is the general solution of a homogeneous differential equation.
(3) Linear Differential Equation :

If p(x) and q(x) are functions of variable x, then the differential equation
Page 2
3
dy
dx
+P(x).y=Q(x) is called a linear differential equation.

Solution :
If we multiply both sides by I.F. =
( ).
p x dx
e

.
We get,
( ). ( ). ( ).
( ) ( )
p x dx p x dx p x dx
dy
e p x ye x e
dx

  
 
( ). ( ).
. ( )
 
 
 
 
 

p x dx p x dx
d
y e x e
dx
( ). ( ).
. ( )
 
 


p x dx p x dx
y e x e
This is the general solution of a linear differential equation.
Application in geometry :
Let y = f(x) is a given curve. Slope of the tangent at the point (x
0
, y
0
) is =
 
0 0
,
 
 
 
x y
dy
dx
.

The equation of the tangent to the curve at point (x
0
, y
0
) is y - y
0=
 
 
0
0
,
 

 
 
o
x y
dy
x x
dx
.

The equation of the normal to the curve at point (x
0
, y
0
) is y - y
0
=
 
 
0 0
0
,
 

 
 
x y
dx
x x
dy
.

Any point,
(1) Length of the tangent
2
1
 

 
 

dy
y
dx
PT
dy
dx
.
(2) Length of the normal
2
dy
PG= y 1+
dx
 
 
 
Page 3
4
(3) Length of subtangent
dy
dx
y
TM=
(4) Length of subnormal
dy
MG= y
dx
Page 4
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QUESTION BANK
(1) The degree of the differential equation is
3 1
2 2
2 1
y - y + 1 = 0 ________.
(A) 6 (B) 3 (C) 2 (D) 4
(2) The order of the differential equation whose general solution is given by
y = c
1
2
x c
e

+ (c
3
+ c
4
) . sin (x + c
5
),
where c
1
, c
2
, c
3
, c
4
, c
5
are arbitrary constant is ________.
(A) 5 (B) 4 (C) 3 (D) 2
(3) The degree of the differential equation of all curves having normal of constant length c
is.
(A) 1 (B) 2 (C) 3 (D) none of these
(4) The degree of the differential equation
3
3 2 2
2
3 2 2
d y d y d y
+ 7 = x . log is
dx dx dx
 
 
 
 
is :
(A) 2 (B) 3
(C) 1 (D) degree doesn’t exist
(5) The degree of the differential equati on satisfying
2 2 2 2
1+x + 1+y = k x 1+y - y 1+x is
 
 
 
:
(A) 4 (B) 3 (C) 1 (D) 2
(6) If m and n are order and degree of the equation

3
2
5
2
2 3
2
2 3 3
3
d y d y
+ 4 = x .- 1
dx dx
 
 
 
 
 

 
 
 
d y
dx
d y
dx
, then :
(A) m = 3, n = 2 (B) m = 3, n = 3 (C) m = 3, n = 5 (D) m = 3, n = 1
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6
(7) The degree and order of the differential equation of the family of all parabolas whose
axis is x-axis, are respectively.
(A) 1, 2 (B) 3, 2 (C) 2, 3 (D) 2, 1
(8) The differential equation representing the family of curves y
2
= 2c
 
cx
, where c is
a positive parameter, is of order and degree as follows.
(A) order 1, degree 1 (B) order 1, degree 2
(C) order 2, degree 2 (D) order 1, degree 3
(9) The differential equation whose solution is Ax
2
+ By
2
= 1, where A and B are arbitrary
constants is of.
(A) second order and second degree (B) first order and first degree
(C) first order and second degree (D) second order and first degree
(10) Order and degree of differential equation of all tangent lines to the parabola y
2
= 4ax is
________ .
(A) 2, 2 (B) 3, 1 (C) 1, 2 (D) 4, 1
(11) The order of differential equation of all parabola with it’s axis paralled to y-axis and
touch x-axis is.
(A) 2 (B) 3 (C) 1 (D) none of these
(12) Which of the following differential equation has the same order and degree ________ .
(A)
6
4
x
4
d y dy
+ 8 + 5y = e
dx
dx
 
 
 
(B)
2
3
8
3
d y dy
5 + 8 1 + 5y = x
dx
dx
 
 

 
 
 
 
 
(C)
2
2
dy dy
y = x + 1+
dx dx
 
 
 
(D)
2
3
3
3
3
dy d y
1 = 4
dx
dx
 
 

 
 
 
 
 
(13) The differential equation of all conics having centre at the origin is of order.
(A) 2 (B) 3 (C) 4 (D) 5
(14) The order of the differential equation of family of circle touching a fixed straight line
passing through origin is.
(A) 2 (B) 3 (C) 4 (D) none of these
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(15) The order and degree of the differential equation
3
2
2
2
2
2
1
y =
d y
dx
dy
dx
 
 

 
 
 
 
 
are (respectively)
(A) 2, 1 (B) 2, 2 (C) 2, 3 (D) 2, 6
(16) Which of the following equations is a linear equation of order 3 ?
(A)
3 2
3 2
d y d y dy
+ . + y = x
dx
dx dx
(B)
3 2
2 2
3 2
d y d y
+ + y = x
dx dx
(C)
3 3
x
3 3
d y d y
x. + = e
dx dx
(D)
2
2
d y dy
+ = log x
dx
dx
(17) Integrating factor of differential equation
1 dy 1
. + y = 1
cosx dx sinx
is.
(A) sec x (B) cos x (C) tan x (D) sin x
(18) The integrating factor of the differential equation
dy
.(x log x) + y = 2log x is :
dx
(A) e
x
(B) log x (C) log(logx) (D) x
(19) Integrating factor of differential equation
1
- logx
x
2
dy
x + ylog x = x. e . x ; x o is :
dx

(A) x
log x
(B)
 
 
2
log
x
e
(C) e
x
2
(D)
log
x
x
(20) If sin x is an Integrating factor of
dy
p.y Q
dx
 
then p is :
(A) sin x (B) log sin x (C) cot x (D) log cosx
Page 7
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(21) Integrating factor of differential equation
dy
(1+x) - x. y = 1 - x
dx
is :
(A) 1 + x (B) log (1 + x) (C) e
-x
(1 + x) (D) x . e
x
(22) The order and degree of differential equation
2 2
1 1y dx x dy o   
is ________ .
(A) order 1, degree 1 (B) order 1, degree 2
(C) order 2, degree 1 (D) order and degree doesn’t exist
(23) The degree of differential equation
2 3
2 1
( )y y y 
is ________ .
(A)
1
2
(B) 2 (C) 3 (D) 4
(24) The order and degree of the differential equation
2
3
3
3
1 3 4.
dy d y
dx
dx
 
 
 
 
are
(respectively) ________ .
(A) 1,
2
3
(B) 3, 1 (C) 3, 3 (D) 1, 2
(25) The Integrating factor of the differential equation
2
(1 ) 1
dx
y yx
dy
  
is :
(A)
2
1
1 y
(B)
2
1 y
(C)
2
1
1 y
(D)
2
1 y
(26) y
2
= (x - c)
3
is general solution of the differential equation : (where c is arbitrary constant).
(A)
3
27
dy
y
dx
 

 
 
(B)
3
2 8 0
dy
y
dx
 
 
 
 
(C)
3
8 27
dy
y
dx
 

 
 
(D)
3
3
8 27 0
d y
y
dx
 
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(27)
y = ae
2x
+ be
-3x
is general solution of differential equation :
(A)
2
2
6
d y dy
y
dx
dx
 
(B)
2
2
6
d y dy
x y
dx
dx
 
(C)
2
2
0
d y dy
y
dx
dx
  
(D)
2
2
0
d y dy
x y
dx
dx
  
(28) The differential equation of family of curves y = Ax +
 
 
 
B
x
is :
(A)
2
2
2
d y dy
y x y o
dx
dx
  
(B)
2
2
2
d y dy
y x y o
dx
dx
  
(C)
2
2
2
0
d y dy
x x y
dx
dx
  
(D)
2
2
2
0
d y dy
x x y
dx
dx
  
(29) Family of curves y = e
x
(A cos x + B sin x) represents the differential equation : ________
. (where A and B are arbitrary constant)
(A)
2
2
2
d y dy
y o
dx
dx
  
(B)
2
2
2 2
d y dy
y o
dx
dx
  
(C)
2
2
2 0
d y dy
y
dx
dx
  
(D)
2
2
2 2 0  
d y dy
y
dx
dx
(30) The differential equation of family of parabolas with focus at origin and x-axis as axis is :
(A)
2
2
dy dy
y x y
dx dx
 
 
 
 
(B)
2
2
dy dy
y xy y
dx dx
 
 
 
 
(C)
2
2
dy dy
y xy y
dx dx
 
 
 
 
(D)
2
2
dy dy
y x y
dx dx
 
 
 
 
(31) The differential equation of all parabolas having the directrix parallel to x-axis :
(A)
3
3
0
d x
dy
(B)
3
3
0
d y
dx

(C)
3 2
3 2
d y d y
o
dx dx
 
(D)
2
2
d y
o
dx

Page 9
10
(32) The differential equation of all parabolas having axis parallel to y-axis :
(A)
3
3
0
d x
dy
(B)
3
3
0
d y
dx

(C)
3 2
3 2
d y d y
o
dx dx
 
(D)
2
2
d y
o
dx

(33) The differential equation of family of hyperbolas with asymptotes x + y = 1 and x - y = 1
is :
(A) yy
1
= x - 1 (B) yy
1
+ x = 0 (C) yy
2
= y
1
(D) y
1
+ xy = 0
(34) The differential equation of family of circles of radius ‘a’ is :
(A) a
2
y
2
= [1 - y
1
3
]
2
(B) a
2
y
2
= [1 - y
1
2
]
3
(C) a
2
(y
2
)
2
= [1 + y
1
3
]
2
(D) a
2
(y
2
)
2
= [1 + y
1
2
]
3
(35) Family y = Ax + A
3
of curves is represented by the differential equation of degree :
(A) 1 (B) 2 (C) 3 (D) 4
(36) The differential equation of all non-vertical lines in a plane is :
(A)
0
dy
dx

(B)
3
3
0
d x
dy
(C)
2
2
0
d y
dx
(D)
0
dx
dy
(37) The differential equation of the family of circles with fixed radius 5 units and centeres
on the line y = 2 is :
(A)
2
2 2
dy
(y-2) = 25-(y-2)
dx
 
 
 
(B)
2
2
dy
(y-2) = 25-(y-2)
dx
 
 
 
(C)
2
2
dy
(x-2) = 25-(y-2)
dx
 
 
 
(D)
2
2 2
dy
(x-2) = 25-(y-2)
dx
 
 
 
(38) The differential equation of all circles passing through the origin and having their centres
on the x-axis is :
(A) y
2
= x
2
+ 2xy
dy
dx
(B) y
2
= x
2
- 2xy
dy
dx
(C) x
2
= y
2
+ xy
dy
dx
(D) x
2
= y
2
+ 3xy
dy
dx
Page 10

Differential Equation (Solved MCQs and Notes)

Differential Equation (Solved MCQs and Notes)

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