DIFFERENTIAL EQUATION
CHAPTER – 9
DIFFERENTIAL EQUATION
MASTER CARD
Points to Remember
1.
a) Definition
b) Order and degree
c) Linear and non-linear
2.
Formation of differential equation
3.
Solution of differential equations:
a) By the method of variable separable
b)
Solving differential equation of the form
c)
Homogeneous differential equation
d)
Solving differential equation of the form
e)
A)
B)
i)
ii)
dy
 f  ax  by  c 
dx
d2 y
 f  x.
dx 2
Solving differential equation of first order and degree one linear Differential equation i.e.
Definition : An equation containing an independent variable and differential coefficients of
dependent variables w.r.t. independent variable is called a differential equation.
Order and Degree :
The order of a differential equation is the order of the highest order derivative appearing in the
equation.
The degree of a differential equation is the degree of the highest order derivative when differentials
are made free from radicals and negative powers.
DIFFERENTIAL EQUATION
Working rule for the formation of Differential Equations
Step I
Step II
Step III
Write the given equation.
Count the number of distinct arbitrary constants present in the given equation.
Differentiate the given equation successively as many times as the number of arbitrary
constants.
Step IV Eliminate the arbitrary constants by using the given equation and equation obtained in the
Step III. The equation obtained is required differential equation.
dy
Working Rule for solving
 f  x  .g  y 
dx
Step I Bring expression involving x on the side and expression involving y on the other side. Always keep
dx and dy in the numerators.
Step II Integrated both sides and add arbitrary constant c only on one side. This gives th required general
solution.
Working Rule for solving
dy
 f  ax  by  c 
dx
Step I Identity the function f(ax + by + c).
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DIFFERENTIAL EQUATION
Step II
Put z = ax + by + c and differentiate w.r.t.x. solve this to find the value of
Step III
Put the value of
Step IV
dy
.
dx
dy
and ax + by + c in the given differential equation. Separate the variables
dx
z and x and integrate both sides.
Reduce the value of Z. This gives the general solutions of the given differential equation.
dy
 f  y / x
dx
Step I Make sure that RHS is either a function of ‘y/x’ or the quotient of two homogeneous function of
‘same’ degree.
dy
dv
vx
Step II
Put y = vx and differentiate it w.r.t.x to get
dx
dx
dy
Step III Put the value of
and y in the given differential equation. Separate the variables v and x
dx
integrate both sides.
Step IV Replace the value of v. This gives the general solution of the given differential equation.
Working Rule for solving
Step I
Step II
dy
+ Py = Q
dx
Identify P and Q and sure that these are function of x only.
Evaluate  Pdx
Step III
Step IV
Find e  . This is the integration factor (I.F)
Put the values of I.F. in the general solution y(I.F.) =  Q  I .F .  dx + c and simplify it. This
Working Rule for solving
Pdx
gives the general solution of the differential equation.
Working Rule for solving
d2 y
 f  x
dx 2
dy
 f  x  dx  c
1
dx 
Step I
Integrate the given equation w.r.t.x. and get
Step II
Simplify
Step III
Integrate the equation (1) w.r.t.x and get y
Step IV
Put the values of I.F. in the general solution y  I .F .    Q  I .F .  dx  c and simplify it. This
 f  x  dx in 1
 f  x  dx  c )dx  c
1
2
. This is the required solution.
gives the general solution of the differential equation.
QUESTIONS :
A)
Definition
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2
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DIFFERENTIAL EQUATION
1.
Which of the following is differential equation
dy
a)
 x .log x
dx
b)
y.
dy
4 x
dx
2.
B)
Order and Degree
Find the order of the following differential equations :
3.
d2s
1d 3 y
 ds 
a)
 3   4  0
b)
 ex
2
3
dt
xdx
 dt 
Find the degree of the following differential equations :
3
2
a)
4.
d2 y
 dy 

4
x

 dx 
dx 2
 
dy
 4 sin x
dx
b)
C)
Linear and non-linear
Find whether the following differential equations are linear or non-linear :
dy
dy
d2 y
a)
 4 sin x
b)
 2  y. 3
dx
dx
dx
6.
D)
Solving of differential equation
Find the differential equation for the family of curves given by y = A x + B/x where A, B are
arbitrary constants.
Solve the differential equation  x 2  yx 2  dy   y 2  x 2 . y 2  dx  0
7.
Solve the differential equation
5.
8.
9.
10.
1.
dx
2
  4 x  y  1
dy
dy
 x 2  xy  y 2
Solve the differential equation x 2
dx
dy
 y  tan x .
Solve cos 2 x
dx
d2x
dx
 1  sin y , given that x = 0, x  0,
Solve the differential equation
 0 when y  0.
2
dy
dy
GRADED QUESTIONS
Each question carry 3 marks
LEVEL - 1
Find differential equation of the following family of curve.
B
Y  Ax 
where A and B are arbitrary constants.
x
2.

Ans:
d2 y
dy
x
x
 y0
2
dx
dx
2

Find the differential equation of the family of circles x  a 2   y  b   r 2 by eliminating a and b.
2
3
2
  dy 2 
2d y
Ans : r  2   1      0
 dx    dx  
2
3.
Show that y  a .e 2 x  be  x is a solution of y2  y1  2 y  0
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3
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DIFFERENTIAL EQUATION
4.
5.
6.
8.
9.
10.
1.
d2 y
dy
Where y2  2 and y1 
.
dx
dx
Solve the differential equation
dy
 1  x  y  xy
dx
Solve the following initial value problem
dy
 e  y cos x , y  0   0
dx
Solve the differential equation
 dy 
sin 1    x  y
 dx 
dy
 x 2  xy  y 2
. Solve x 2 .
dx
dy
y
 y  x .tan
Solve x .
dx
x
dy
 y, y  0
Solve  x  3 y 2  .
dx
dy
Solve cos 2 x .  y  tan x
dx
Ans: tan  x  y   sec  x  y   x  c 7
y
 log x  c
x
y
Ans: x sin  c
x
y
Ans:
 3y  c
x
Ans: tan 1
Ans:
4.
5.
6.
7.
8.
y  tan x  1  ce  tan x
 d2s 
 ds 
b)
Order:2, Degree : 2, Non-linear
 2   3   4  0
 dt 
 dt 
Find the differential equation for the following family of curve given by y  e x  a cos x  b sin x  .
3
where a and b are arbitraty constants.
3.
x2
c
2
LEVEL – 2
Determine the order and degree of the following differential equations. State, if these is linear or
nonlinear.
1d 2 y
a)
 ex
Order : 2, Degree : 1, Linear
2
xdx
2
2.
Ans: log 1  y  x 
Ans:
d2 y
dy
 2  2y  0
2
dx
dx
d2 y
 1.
dx 2
d2 y
 x 2 y  0.
Show that y  a cos x  b sin x in solutin of
2
dx
Show that y  e x  ax  b in solution of e x
x
 xc
2
y2
Ans: log xy  x 
c
Solve the differential equation  y  xy  dx  xd  xy 2 dy  0
2
y
Ans:  log( x )  c
Solve the differential equation sec2 x2 tan ydx  sec2 y .tan xdy  0.
x
dy
2
 4x  y 1
  4 x  y  1
Ans: tan 1 
Solve the differential equation
  2x  c
dx
2


Solve the following differential equation 1  cos x  dy  1  cos x  dx.
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
4
Ans : y  2 tan

Written By:- Raj Kumar Badhan
DIFFERENTIAL EQUATION
dy
 x  y.
dx
9.
Solve the differential equation x .
10.
Solve the differential equation
1.
LEVEL – 3
Find the differential equation of the following family of curve.
y  a .e  b.e
x
2x
 c .e
3 x
dy
 2 y  e3 x
dx
3.
Solve x 2  cx 2 dy  y 2  x 2 y 2 dx  0
4.
Solve  x  2 
5.
6.
7.
8.
9.


Ans:
dy
 4. x 2 . y
dx
Solve cos x 1  cos y  dx  sin y 1  sin x  dy  0
Solve the differential equation
dy
 cot 2  x  y 
dx
Solve the differential equation
 x 2  y 2  dx  2 xy dy  0 given that y  1 when x  1
1 1
  log y  x  c  0
x y
Ans : log y  2 x 2  8 x  16 log x  2  c
Ans: 1  sin x 1  cos y   c
Ans:
2 y  2 x  sin 2  x  y   c
Ans:
Solve the differential equation
d2 y
dy
 sin x given that
 1 and y  1 when x  0
2
dx
dx
Solve the differential equation
d2x
dy
x  2 y2 .
 y , y  0 given that y  1 when x  2
2
dx
dy
Solve the differential equation
d2x
dx
 1  sin y given that x  0,
 0 when y  0
2
dy
dy

10.
d3 y
dy
Ans:
7
 6y  0
3
dx
dx
b
x2 y
dy
is a solution of x 2 . 2  x .  y  0
x
dx
dx
Show that y  ax 

y
 log  x   c
x
Ans : y  e 3 x  ce 2 x
where a , b, c are artibitrary constants.
2.

Ans:
x2  y2  2x
Ans: y 

1 3
x  sin x  1
6
Ans: x  2 y 2
Ans:
x
y2
 sin y  y
2
Important Questions
x
d2 y
 ax  b is a solution of the differential equation e
1
dx 2
1.
Show that y  e
2.
Show that the differential equation of which y  2  x 2  1  ce  x 2 is a solution is
3.
4.
x
dy
 2 xy  4 x 2
dx
d2 y
dy
 6  9 y  0.
Show that y   A  Bx  e is a solution of the differential equation
2
dx
dx
Solve the following differential equations.
dy
x.
 y  x2  y2
dx
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3x
5
Written By:- Raj Kumar Badhan
DIFFERENTIAL EQUATION
9.
dy
 y
 y  x .tan    tan 1 x
dx
 x
dy
 y  tan 1 x .
1  x 2  dx
dy
cos x
 cos 2 x  cos 3 x
dx
dy
 x  2 log  1 , given that y  0, when x  2
Solve
dx
Find the particular solution of 1  x 2  dy  x 1  y 2  dx  0 given that y = 1 when x = 0
10.
Solve
5.
6.
7.
8.
11.
12.
13.
14.
15.
x
dy

 y tan x  2 xx 2 tan x
0 x
dx
2
Solve the differential equation
d2 y
dy
 sin x  x given that
 1 when x  0
2
dx
dx
d2 y
dy
 x 2 sin x subject to the conditins that
 1 when x  0
Solve
2
dx
dx
d2 y
 sin 2 x  cos 3 x .
dx 2
d2 y
 log x
dx 2
d2 y
 2x  x2  e2 x .
2
dx
QUIZ
(Differential Equation)
 d2 y 
d2 y
 dy 
 3    x log  2  is
The degree of differential equation
dx 2
 dx 
 dx 
a)
1
b)
2
c)
3
d)
none of these
The order of the differentia equation of all tangent lines to the parabola y  x 2 is
a)
1
b)
2
c)
3
d)
4
2
1.
2.
3
3.
4.
5.
  dy 2  2
 d2 y 
The differential equation 1      R  2  is of the
 dx 
  dx  
a)
first order and second degree
b)
second order & second degree
c)
third order and second degree
d)
second order & third degree
2
d y
 0 is that of a
The differential equation
dx 2
a)
family strainght lines
b)
family of circles
c)
family of parabolas
d)
none of these
The differential equation  x  2 y  dx  dy   dx  dy is of type
a)
c)
variable seperable
homogeneous
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b)
d)
6
reducible to variably seperable
linear differentia equation-
Written By:- Raj Kumar Badhan
DIFFERENTIAL EQUATION
6.
7.
8.
9.
10.
11.
12.
13.
The differential equation  x 3 y 3  xy  dy  dx is a
a)
variable seperable
b)
linear in y
c)
educible to homogeneous
d)
linear differential equation in x
The solution of the differential equation xdy  ydex  0 is given by
a)
y = kx
b)
x+y=k
c)
xy = k
d)
x–y=k
The differential equation of all parabolas with axis parallel to the axis of y is
y23  y1
a)
y2 = 2y1 + x b)
y3 = 2y1
c)
d)
none of these
y 2  2 xy  x
passing through (1, –1)
y 2  2 xy  x 2
a)
straight line b)
circle
c)
ellipse
d)
none of these
Equation of curve through the origin satisfying dy = (sec x + y tan x)dx is
a)
y sin x = x
b)
y cos x = x
c)
y tan x
d)
none of these
2
dy x
The general solution of

is
dx y 2
1
a)
y  e x  ce 2 x
b)
x3  y3  c
c)
x2  y2  c d )
x2  y2  c
3
dy x 2
The general solution of
is

dx y 2
The curve satisfying the equation y1 
a)
x3  y3
b)
x3  y3  c
c)
Which of the following is not a differential equation ?
d
a)
ax 2  bx  c   y
b)

dx
d
c)
d)
 x  y  c
dx
x2  y2  c
d)
x2  y2  c
dy
 ax 2  bx  c
dx
d
 sin y   x
dx
ASSIGNMENT
(Differential Equation)
Form the differential equation for the following curves on 1 to 3.
1
1.
y  ke sin  3
2.
y  ae x  be 2 x  ce 3 x
3.
y2  a b  x2
4.
Show that y  ae 2 x  be  x is solution of y2  y1  2 y  0


Solve the following differential equation :
5.
6.
7.
8.


3e x tan ydx  1  e x sec 2 y dy  0
dy
 e x y  x 3e  y
dx
dy
x2
 x 2  xy  y 2
dx
dy
 2 y  e3 x
dx
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7
Written By:- Raj Kumar Badhan
DIFFERENTIAL EQUATION
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
(1 + y²)dx + xdy = 0 given that y(1) = 1
x log xdy – ydx = 0
dy
x  e 2 y  1   x 2  1 e y
 0.
dx
dy
dy
y2  x2
 xy
dx
dx
dy
x  y 1

dx 2 x  2 y  1
dy
4x
1
 2

2
dx x  1
x2  1


dy y
 y
  tan  
dx x
x
dy
1
 2 xy  2
 x 2  1 dx
x 1
2
3
3
x ydx   x  y  dy  0
d2 y
dy
 x 2 sin x given that
 0 and y  0 when x  0
2
dx
dx
dy y
y
2
 log y  2  log y 
dx x
x
EVALUATION
(Differential Equation)
a
and y 2  4ax satisfy the differential equation
c
1.
Verify that both y  cx 
2.
dy
dx
a
dx
dy
Find the differential equation of the family of curve :
y = cos (x + m)
Find the differential equation of the family of curve:
(x + a)²– 2y² = a²
yx
3.
4.
5.
6.
7.
8.
Solve the following differential equation :
dy
 1  x  y  xy
dx
dy
1
ex
 3 y  2  given that y  0  
dx
2
dy
 cos  x  y 
dx
1
dy
 y  e tan x
1  x 2  dx


ydx  x  y 3 dy  0
Graphics By:- Pradeep
8
Written By:- Raj Kumar Badhan
DIFFERENTIAL EQUATION
9.
10.
11.
1  y  dx  xdy  0 given that y 1  1
2
dy
 y  x2  y2
dx
2
d x
 x cos x
dy 2
x
12.
d2x
dx
 1  sin y given that x  0,
 0 when y  0
2
dy
dy
13.
e dx  x 1  given that x  0, y  3
dy
Graphics By:- Pradeep
9
Written By:- Raj Kumar Badhan