1

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS
7.1

BASIC CONCEPTS
(a)

An equation involving derivative(s) of the dependent variable with respect to only one
independent variable is called an ordinary differential equation. In particular equation involving

dy
or dx, dy along with the variables x and y is known as an ordinary
dx

differential equation in x and y.

(b)

Order of a differential equation is defined as the order of the highest order derivative of
dependent variable with respect to the independet variable, involved in the given
differential equation.

(c)

Degree of a differential equation, when it is polynomial equation in derivatives, is

defined as the highest power (exponent) of the highest order derivative involved in the
given differential equation.

(d)

Solution of a differential equation is a function, in the variables involved in the

differential equation, which satisfies the given different equation.

(e)

The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution (primitive) of the differential equation.

(f)

The solution free from arbitrary constants, i.e., the solution obtained from the general
solution by giving particular values to arbitrary constants is called a particular solution of the differential equation.

7.2

FORMATION OF A DIFFERENTIAL EQUATION

(a)

Find how many arbitrary constants are given in the function and how many we are
asked to eliminate.

(b)

If we have to eliminate one arbitrary constant, then we can differentiate the function
once ; if two then we can differentiate twice and so on.

(c)

Then from the given function and the expression obtained after differentiation, eliminate the arbitrary constants, to get the differential equation.

7.3

SOLUTION OF DIFFERENTIAL EQUATION

(a)

In the solution of a differential equation by separating the variable method, take

involving x with dx on one side and terms involving y with dy on the other side and
integrate both sides. dx and dy should always be in numerator.

(b)

Homogeneous differential equation are of the type

(x 2 + xy) dx (y2 + 2xy)dy = 0,

dy x + y
dy x 2 + xy
=
or dx = x - y or
dx
y2

2
2
y
dy y
dy x + y - x
=
=
+ sin , etc, i.e,
or
dx x
x
dx
x

combined degree of each term involved is same or

dy
y
= f . We also say that the
dx
z

homogeneous function involved is of degree zero. An equation is a homogeneous

equation, if for function involved f(lx, ly) = f (x, y).

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DIFFERENTIAL EQUATIONS
In such cases, we substitute

dv
dy
y
= v or y = vx
= v+x
and proceed by substituting for
dx
dx
x

dy
and y in the given equation. Sometimes, we might get the homogeneous equation as
dx

x
dx
dx
dv
= f . In such case, we substitute x = vy
y
and proceed by substituting
=
v
+
dy
dy
dy
y
dx
for dy and x in the given equation.

(c)

In linear differential equation of first order

dy
+ P( x ) . y = Q(x) ; P(x), Q(x) are
dx

function of x.
Find Integrating Factor (I.F.) = e P dx and use the result y. (I.F.) =

{(I.F} Q(x)} dx as solution.

dx
Sometimes, we get linear differnetial equation form as dy + P( y).x = Q( y );P( y ), Q( y ) are funcitons

of y. In such cases find I.F. = e P dy and use the result x. (I.F.) =

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{(I.F.) Q( y)} dy as solution.

DIFFERENTIAL EQUATIONS

SOLVED PROBLESM
Ex.1

Write the order and degree of the differential equation

Sol.

Highest order derivative is

dy
dy
+ sin
=0
dx
dx

dy
dx

\
Order of differential equation is 1.
Equation canot be written as a plynomial in derivatives. Hence degre is not defined.
Ex.2

What will be the order of the differential equatin, corresponding to the family of curves

Sol.

y = a sin (x + b), where a is arbitrary constant.

As there is one arbitrary constant, so order of corresponding differential equation is 1.

Ex.3

Form the different equation representing the family of curves given by xy = Aex + Be
where A and B are constants.
xy = Aex + Bex
x

Sol.

...(i)

y + xy = Ae Be and y + y + xy" = xy [From(i)]

equation
x

Ex.4

Find the equation of a cure passing through the point (2, 3), given that the slope of the
tangent to the curve at any point (x, y) is

Sol.

2x
.
y2

We know that the slope of the tangent to a curve at any point (x, y) is given by

So,

dy
.
dx

dy 2 x
=
Integrating both sides, we have
dx y 2

y dy = 2x dx
2

y3
= x2 + C
3

...(1)

Since the curve (1) passes through the point (2, 3)

Substituting this value of C in (1), we have

Ex.5

xy' + 2y xy = 0 is the required

(3)3
= ( -2)2 + C 9 = 4 + C C = 5
3

y3
= x2 + 5
3

i.e., y3 3x2 = 15 or y =

3x 2 + 15

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line
segment joining the point of contact to the point (4, 3). Find the equation of the
curve, given that it passes through (2, 1).

Sol.

It is given that

dy
3+ y
-3-y
= 2
= 2

4
x
dx
4+x

1
2
dy =
dx
3+y
4+x

On integrating, we have,
log |3 + y| = 2 log | 4 + x | + C
Since the curve passes through the point (2, 1)

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DIFFERENTIAL EQUATIONS
we substitute the value x = 2 and y = 1 in Eq. (1) and get
log |3 + 1| = 2 log |4 2| + C
C = log 4 2 log 2 = 2 log 2 2 log 2 = 0

Hence, from (1), the required equation of the curve is

log |3 + y| = 2 log |4 + x|
or 3 + y = (4 + x)2

i.e.,

y + 3 = (x + 4)2

Ex.6

A population grows at the rate of 2% per year. How long does it take for the polulation

Sol.

do double ?
Let P0 be the initial population and let the population after t years be P. Then,
dP
2P
=
(given)
dt 100

dP
1
dt Integrating both sides, we have
=
P
50

log P =

1
t+C
50

When t = 0, P = P0

2P
When P = 2P0 t = 50 log 0
P0

P
1
Hence, log =
t
P
50
0

\ log P0 = C

= 50 log 2 = 50 0.3 = 15 years

Thus, it takes 15 years for the population to be double.

Ex.7

y dy
y
= y cos + x is homogeneous and solve
Show that the differential equation x cos
x
dx

x

it.
Sol.

The given differential equation can be written as

y
y cos + x
dy
x
=
dx
y
x cos
x

...(1)

It is a differential equation of the form

dy f ( x, y )
=
dx g( x, y )

y
y
Here, f(x,y) = y cos + x and g(x, y) = x cos Each one is clearly a homogeneous function of
x
x

degree 1.
Put

y = n x or

y
= n ...(2) Differentiating equation (2) with respect to x, we get
x

dn
dy
= n+x
...(3)
dx
dx

Substituting the value of y and

dy
in equation (1), we get
dx

n+x

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dn n cos n + 1
=
dx
cos n

DIFFERENTIAL EQUATIONS
or

dn n cos n + 1
-n
=
dx
cos n

or

dn
1
=
dx cos n

...(4)

Separting the variables in equation (4), we get cos n dn =

dx
x

...(5)

Integrating both sides of equation (5), we get

1

cos ndn = x dx
or

sin n = log |x| + log |C|

or

sin n = log |Cx|

Replacing n by

...(6)

y
y
in equation (6), we get sin = log | Cx |
x
x

or
y = x sin1 {log |Cx|}
which is the general solution of the differential equation (1).
Q.8

Show that each of the following differntial equation is homogeneous and solve it :
x

Sol.

...(7)

dy
y
- y + x sin = 0
dx
x

The differential equation can be written as

dy
=
dx

y
y - x sin
x
x

f (x, y) is a homogeneous function of degree 1.

homogeneous function of order 1.
Thus, the given equation is a homogeneous differential equation.
Put y = nx so that

dn
dy
= n+x
dx
dx

Substituting the vaues of y and

dn
n+x
=
dx

dy
in (1), we have
dx

nx
nx - x sin
dn
dx
1
x = n - sin n x
= sin n
.dn = dx
x
sin n
x

Integrating both sides, we have

1

sin n dn = -

tan

dx

cos ec ndn = - log x+ | log C |

n
C
y
C
=
=
tan
2
x
2x x

i.e., x tan

tan tan

n
= - log x + | log C |
2

y
=C
2x

Ex.9

Show that the equation of the curve whose slope at any point (x, y) is equal to y + 2x
and which passes through the origin by y + 2(x + 1) = 2ex

Sol.

Here, we have
dy
dy
= y + 2x
- y = 2x
dx
dx

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DIFFERENTIAL EQUATIONS
Here, P = 1 and Q = 2x
\ I.F. = e

-1 1 .dx
=e

P dx

= e -x =

1
ex

So, the general solution of the differential equation (1) is

y.

1
2x
1
=
dx + C y. x = 2 x.e - x dx + C = 2[ -xe - x + e - x dx] + C
ex
ex
e

yex = 2xex 2ex + C

Since the curve passes through the origin (0, 0) we have

0e -0 = -2.0.e -0 - 2e -0 + C

Hence, the equation of the curve is

yex = 2xex 2ex + 2
i.e.,

C=2
y = 2x 2 + 2ex

or

y + 2(x + 1) = 2ex

Ex.10 Find the equation of a curve passing through the point (0, 2) given that the sum of the
coordinates of any point on the curve exceeds the magnitude of the slope of the tanSol.

gent to the curve at that point by 5.

According to the question,
x+y=

dy
+5
dx

Here, P = 1 and Q = x 5
I.F. = e

P dx

=e

- 1dx

...(1)
So, the general solution of the D.E. (1) is

= e- x

ye - x = ( x - 5)e - x dx + C

dy
y=x5
dx

ye - x = -xe - x - e - x + 5e - x + C

Since the curve passes thorugh (0, 2),

ye - x = xe - x dx - 5e - x dx + C

y = x 1 + 5 + Cex
2 = 0 1 + 5 + Ce0

C=2

Ex.11 Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate (abscissa)
and the product of the x-coordinate and y - coordinate (ordinate) of that point.
Sol.

According to the question,

dy
= x + xy
dx

dy
- xy = x
dx

............(1)
x2

- xdx
P dx
I.F. = e
= e
=e 2

Here, P = x and Q = x

x2

x2

So, the general solution of the D.E. (1) is ye - 2 = xe - 2 dx + C

Let

I = xe

x2
2

Hence, from (2)

dx

Put -

ye

x2
2

x2
= t so that x dx = dt
2

= -e

x2
2

............(2)

I = - e t dt = -e t = -e

x2
2

x2

+C

i.e., y = -1 + Ce 2

Since the curve (3) passes through (0, 1),

1 = 1 + Ce0
C=2
So, the required equation of the curve is

............(3)
y + 1 = 2e

x2
2

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DIFFERENTIAL EQUATIONS

UNSOLVED PROBLEMS
EXERCISE I
Q.1

Determine the order and egree of each of the following differential equation
dy
(i) x4 4xy 3x3 = 0
dx

(ii)

1 d2 y
+ 9 y = - 4e - x
2
2
x dx

(iii) xy

dy 1 + y 2
=
dx 1 + x 2

(1 + x + x 2 )

Q.2

Form the differential equation corresponding to y2 2ay + x2 = a2 by eliminating a.

Q.3

Form the differential equation of the equation (x + a)2 2y2 = a2 by eliminating a.

Q.4

Form the differential equation corresponding to (x a)2 + 2y2 = a2 by eliminating a.

Q.5

Form the differential equation y = ax2 + bx + c. eliminating a, b, c.

Q.6

Show that y = A sin x + B cos ex + x sin x is a solution of the differential equation y +

d2 y
dx2

2e cos x

x 2 + a 2 )2 satisfies the differential equation (a2 + x2)

d2 y

dy
=0
dx

Q.7

Show that y = log (x +

Q.8

Show that y = ce tan

Q.9

Show that y = Ax +

Q.10

Solve the differential equation

Q.11

Solve the differential equation cos x

Q.12

Solve the differential equation (1 + x2)

Q.13

Solve the differential equation cos y dy + cos x sin y dx = 0 given that y =

Q.14

Solve the differential equation e dx = x + 1, given that y = 3 when x = 0.

Q.15

Solve the differential equation (1 + y2) dx xy dy = 0, and which passes through (1, 0).

Q.16

Solve the differential equation x

Q.17

Solve the differential equation xe y/x y sin

-1

is a solution of the differential equation (1 + x)2

d2 y
dx

dx2

+x

+ (2x 1)

dy
=0
dx

B
dy
d2 y
, x 0 is a solution of the differential equation x 2
+x
-y=0
x
dx
dx
1
dy
= cos3 x sin2 x + x 2x + 1 , x - 2 ,

dx

dy
cos 2x = cos 3x
dx
dy
x = 2 tan1 x
dx
p
p
when x =
2
2

dy

y
y
dy
p
sin + x y sin = 0, y (1) =
x
dx

x
2

y
p
dy
+x
sin = 0, y(1) = 0
2
dx
x

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DIFFERENTIAL EQUATIONS
x
dx +
y

2
x
y - x 2 log dy = 0, given that y (1) = e
y

Q.18

Solve the differential equation xy log

Q.19

Solve the differential equation

Q.20

Solve the differential equation (1 + y 2) dx = (tan1 y x)dx, y(0) = 0

Q.21

Solve the differential equation x

Q.22

Q.23

dy
+ y cot x = 2x + x2 cot x (x 0) given that y (p/2) = 0
dx

dy
y = log x, y(1) = 0
dx

y
y
p
The slope o the tangent at (x, y) to a curve passing through 1, is given by
cos2 .
x
x
4
Find the equation of the curve.

The surface area of a balloon being inflated changes at a rate proportional to time t. If
initially its radius is 1 unit and after 1 second it is 3 units, find the radius after t seconds.

Q.24

The popluation grows at the rate of 5% per year. How long does it take four the population to
duble ?

Q.25

A radioactive substance disintegrates at a rate proportional to the amount of substance

present. If 50% of the given amount disintegrates in 1600 years. What precentage of the
substance disintegrates in 10 years ? [Take elog 2/160 = 0.9957]

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DIFFERENTIAL EQUATIONS

BOARD PROBLES
EXERCISE II
dy
= y(log y log x 1)
dx

Q.1

Solve the differential equation x

Q.2

Solve the differential equation

Q.3

Solve the differential equation (x3 + y3)dy x2ydx = 0

[C.B.S.E. 2002]

Q.4

Solve the differential equation (y + xy)dx + (x xy2) dy = 0

[C.B.S.E. 2002]

Q.5

Show that the differential equation of y2 = 4a(x b) is y

Q.6

Solve the differential equation x

Q.7

Find the differential equation of the family of curves given by x2 + y2 = 2ax.[C.B.S.E. 2003]

Q.8

Solve the differential equation cos 2 x

Q.9

Form the differential equation of xy = Aex + Bex + x2 where A and B are constant

[C.B.S.E. 2001]

dy
= 1 - x + y - xy
dx

[C.B.S.E. 2002]

d2 y dy
+ = 0
dx 2 dx

dy
+ y = x 3 when y(2) = 1
dx

dy
+ y = tan x
dx

[C.B.S.E. 2003]
[C.B.S.E. 2003]

[C.B.S.E. 2003]

[C.B.S.E. 2003]
Q.10

Form the differential equation of curve y2 2ay + x2 = a2, where a is an arbitrary constant.
[C.B.S.E. 2004]
dy
+ 2y tan x = sin x
dx

Q.11

Solve the differential equation

Q.12

Solve the differential equation (x 2 yx2) dy + (y2 + x2y2) dx = 0.

Q.13

2
Solve the differential equation 2x

Q.14

Solve the differential equation

Q.15

Solve the differential equation (x2 + xy) dy + (x2 + y2) dx

Q.16

Solve the differential equation (1 + x2)

Q.17

Solve the differential equation

Q.18

Solve the differential equation 2xy dx + (x2 + 2y2) dy = 0

Q.19

Form the differential equation of y = a sin (x + b), where a and b are arbitrary consants.

Q.20

Form the differential equation of y = A cos 2x + B sin 2x when A, B are constant

dy
- 2xy + y 2 = 0 when y(e) = e.
dx

dy y
+ = ex, x > 0
dx x

dy
- 2xy = (x2 + 2) (x2 + 1)
dx

dy
+ secx. y = tan x
dx

[C.B.S.E. 2004]
[C.B.S.E. 2004]
[C.B.S.E. 2005]

[C.B.S.E. 2005]

[C.B.S.E. 2005]
[C.B.S.E. 2005]

[C.B.S.E. 2006]
[C.B.S.E. 2006]

[C.B.S.E. 2007]

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10

DIFFERENTIAL EQUATIONS

Q.21

Solve the differential equation (x cos y)dy = ex(x log x + 1)dx

Q.22

Solve the differential equation

Q.23

Solve the differential equation x2

Q.24

Form the differential equation for the circles passing through origin and with centres on y-

dy
+ 2y tan x = sin x
dx

[C.B.S.E. 2007]
[C.B.S.E. 2008]

dy
+ 2y tan x = sin x
dx

axis.

[C.B.S.E. 2008]

[C.B.S.E. 2008]
dy
y
= y x tan
dx
x

Q.25

Solve the differential equation x

[C.B.S.E. 2009]

Q.26

Solve the differential equation cos2x

Q.27

Solve the differential equation

Q.28

Solve the differential equation (x 2 + 1)

Q.29

Solve the differential equation (x3 + x2 + x + 1)

Q.30

Solve the differential equation x dy y dx =

Q.31

Solve the differential equation (y + 3x2)

Q.32

Form the differential equation of the family of circles in the second. quadrant and touching

dy
+ y = tan x
dx

[C.B.S.E. 2009]

dy
+ y = cos x sin x
dx
dy
+ 2xy =
dx

x2 + 4

dy
= 2x2 + x
dx

x2 + y2 dx

dx
=x
dy

the coodinate axes.

Q.33

[C.B.S.E. 2009]

[C.B.S.E. 2010]

[C.B.S.E. 2010]

[C.B.S.E. 2011]

[C.B.S.E. 2011]

[C.B.S.E. 2012]

Find the particular solution of the differential equation x(x 2 1)

dy
= 1; y = 0 when x = 2.
dx

[C.B.S.E. 2012]
Q.34

Solve the following differential equation :

Q.35

(`1 + x ) dy + 2 xy dx = cot x dx; x 0

Find the particular solution of the differential equation

[C.B.S.E. 2012]
2

(tan y x) dy = (1 + y ) dx, given that when x = 0, y = 0.

1

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[C.B.S.E. 2013]

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DIFFERENTIAL EQUATIONS

ANSWER KEY
EXERCISE 1 (UNSOLVED PROBLEMS)
1. (i) order-1, degree-1
2
2 dy
2. (x 2y )
dx

5.

d3 y
dx

=0

(ii) order-2, degree-1

4xy

(iii) order-1, degree-1

dy
dy
2
2
2
x = 0 3. x + 2y = 4xy
dx
dx

10. y =

4. 4xy

5
3
1
1
1
1
1
3
5
( 2x + 1) 2 - (2x + 1) 2 + C, x - ,
sin x sin x +
2
6
10
3
5

11. y = sin 2x x + 2 sin x + log |sec x + tan x| + C, x (2n + 1)

1
2
1
2
log |1 + x | + (tan x) + C
2

12. y =

15. x = (1 + y )

18.

x2
2y

log

dy
2
2
= 2y x
dx

13. log sin y + sin x = 1

19. y = x

14. y = (x + 1) log |x + 1| x + 3

y
y
17. e- y/ x sin + cos = 1 + log x 2, x 0
x
x

y
16. log | x | = cos , x 0
x

x2
x
3
log y = 1

y
4y2
4e2

p
,nZ
2

p2
1
tan 1
(sin x 0) 20. (x tan y + 1)e
y=1
4 sin x

21. y = x 1 log x, x > 0

22. y = x tan

24. 20(log 2) years

25. 0.43%

e
log
x

23.

8t 2 + 1

EXERCISE 2 (BOARD PROBLEMS)

1. y = xe

5. xy =

cx

2. log(1 + y) = x

- x3
x2
y2
+ c 3.
=c
3 + log y = c 4. log x + x + log y
3y
2
2

dy
x4
2
2
2 7. x + 2xy
y =0
dx
4

tan x

8. y = tan x 1 + ce

dy
dy

2 dy
- 1 2y2 2xy
10. x
=0
dx
dx
dx

13. y =

2x
1 + log x
2

14. y =

16. y = (x + 1)(x + tan

2

18. 3x y + 2y = c

19.

11. y = cos x + c cos x

1
c
x
(x 1)e +
x
x

9. x

d2 y
dx

+2

12. log y +

dy
2
= xy x + 2
dx

1
1
x=c
+
y
x

y y

15. log x + c = 2 log 1 -

x x

x) + c(x + 1) 17. y(sec x + tan x) = sec x + tan x

d2 y
dx2

+y=0

20.

d2 y
dx2

+ 4y = 0

21. sin y = e log x + c

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12

DIFFERENTIAL EQUATIONS
2

22. y = cos x + cos x

25. log x sin x = c

x

31. y = 3x + cx
33.

y=

23. y =

1
x (x + y)
2

26. y = (tan x 1) + ce
32.

tan x

24. (x y )

30. y +

dy
= 2xy
dx

x2 + y2 = cx

(x + y)2 |(y)2 + 1| = (x + yy)2

3
1 1
1

log1 - 2 - log
4
2
x 2

34. y = (1 + x2)1 log |sin x | + c (1 + x2)1

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