1. By: Prof. Yogiraj Mahajan
HOD Science Dept.
K. K. Wagh Polytechnic, Nashik-3

2. Differential Equations [ Marks-20]
• Differential equation- Definition, order and
degree of a differential equation. Formation of
differential equation containing single
constant.
• Solution of differential equation of first order
and first degree for following types
 Variable separable form .
 Equation reducible to variable separable
form.
 Linear differential equation.
 Homogeneous differential equation.
 Exact differential equation. 2Prof. Yogiraj Mahajan

3. Differential Equations
Definition : An equation consists of
independent variable (x), dependent
variable (y) and differential coefficient
is called differential
equation.
....,,,
3
3
2
2
dx
yd
dx
yd
dx
dy
3Prof. Yogiraj Mahajan

4. Differential Equations
0y
dx
dy
x
2
dx
dy
dx
dy
xy
0
2
2
2
xy
dx
dy
dx
yd
xy
dx
dy
yx 2
22
3
22
2
2
2
1
dx
dy
dx
yd
r
3
2
2
dx
dy
y
dx
yd
3
dx
dy
y
dx
dy
dx
dy
x 5
3
2
2
dx
dy
y
dx
yd
4Prof. Yogiraj Mahajan

5. Differential Equations
Order of the highest order derivative
appearing in differential equation is called
Order of differential equation.
Degree of the highest order derivative
when the derivatives are free from radical
and fraction(-ve index) is called degree
of a differential equation.
Order of differential equation :
Degree of a differential equation:
5Prof. Yogiraj Mahajan

6. Differential Equations
2
dx
dy
dx
dy
xy
0
2
2
2
xy
dx
dy
dx
yd
xy
dx
dy
yx 2
22
3
2
2
dx
dy
y
dx
yd
Order – 1 , Degree – 1
Order – 2 , Degree – 2
Order – 2 , Degree – 1
Order – 1 , Degree – 2
6Prof. Yogiraj Mahajan

7. Differential Equations
0y
dx
dy
x
3
22
2
2
2
1
dx
dy
dx
yd
r
3
2
2
dx
dy
y
dx
yd
3
dx
dy
y
dx
dy
dx
dy
x 5
Order – 1 , Degree – 4
Order – 2 , Degree – 2
Order – 2 , Degree – 2
Order – 1 , Degree – 1
7Prof. Yogiraj Mahajan

8. Differential Equations
Formation of Differential Equation:
The process of eliminating arbitrary
constant from the given relation by
differentiation is called formation of
differential equation.
8Prof. Yogiraj Mahajan

9. Form a differential equation whose solution is y = mx2
2
mxy
x
y
dx
dy 2
dx
dy
x
m
xm
dx
dy
2
1
2.
2
2
2
1
x
y
dx
dy
x
x
y
m
9Prof. Yogiraj Mahajan

10. Differential Equations
Relation between the dependent variable(y) and
independent variable (x) [without their
differential coefficients] along with arbitrary
constant is called general solution of
differential equation.
The solution obtained from the general solution
by giving particular values to arbitrary
constants occurring in it is called particular
solution of differential equation.
10Prof. Yogiraj Mahajan
Solution of Differential Equation:

11. Since the process of solving of a
differential equation recovers a function
from knowing something about its
derivative, it's not too surprising that we
have to use integrals to solve differential
equations.
Differential Equations
11Prof. Yogiraj Mahajan

12. Differential Equations
Solution of differential equation of first
order and first degree for following types
 Variable separable form.
 Equation reducible to variable
separable form.
 Linear differential equation.
 Homogeneous differential equation.
 Exact differential equation.
12Prof. Yogiraj Mahajan

13. Differential Equations
An equation of the form Mdx + Ndy = 0
or where M and N are
functions of x & y is called as differential
equation of first order and first degree.
0
dx
dy
NM
Differential Equation Of First Order And
First Degree :
13Prof. Yogiraj Mahajan

14. Variable Separable Differential Equation
[ By simple rearrangement separate x & y ]
Mdx + Ndy = 0
Where M = f( x) fun. Of x only &
N = g(y) fun. Of y only
Solution is given by
CNdyMdx
Differential Equations
14Prof. Yogiraj Mahajan

15. 2
3xy
dx
dy
dxx
y
dy
32
Cx
y
Cdxx
y
dy
2
2
3
2
1
3
Variable separable diff. eqn.
CNdyMdx
Differential Equations
15Prof. Yogiraj Mahajan

16. 011 dxydyx dyyxydxyx 111
2
0
2222
dxxyydyyxx 0e
2yx
dyedx
xy
0sec1tan3
2
ydyeydxe
xx
2
3xy
dx
dy
0sin11
122
ydyxdxyx
xdyyydxx
22
coscos
1yxxy
dx
dy
16Prof. Yogiraj Mahajan

17. 01
23
ydxxdyx
0
2222
dxxyydyyxx
0sec1tan3
2
ydyeydxe
xx
xyyx1
dx
dy
xy
0sincoscossin yx
dx
dy
yx
dx
dy
ya
dx
dy
xy
2
3
2
1 x
dxx
y
dy
0
11
22
dy
y
y
dx
x
x
17Prof. Yogiraj Mahajan

18. Prof. Yogiraj Mahajan 18
Equation Reducible to Variable Separable
Form
If M or N contain ax + by + c then by
substitution
z = ax + by + c and
Equation can be simplify to variable
separable differential equation.
Differential Equations
dx
dy
ba
dx
dz

19. Prof. Yogiraj Mahajan 19
2
14 yx
dx
dy
22
a
dx
dy
yx
yx
dx
dy
sin
Solve:
342
12
yx
yx
dx
dy
dx
dy
yx 212cos
2

20. If y & are of first degree and not multiplied
by each other then express equation as
Where P & Q are functions of x only.
Solution is given by
Linear Differential Equation
dx
dy
QPy
dx
dy
CdxQeye
PdxPdx
Differential Equations
Integrating factor IF =
Pdx
e
20Prof. Yogiraj Mahajan

21. 322
1
1
1
4
xx
xy
dx
dy
ecxxy
dx
dy
coscot
1221
3
xxyx
dx
dy
xx
ti
dt
di
2sin106
1cossincos xxxy
dx
dy
xx
xy
dx
dy
x tancos
2
0log2
2
xxy
dx
dy
x
xxy
dx
dy 2
costan
21Prof. Yogiraj Mahajan

22. If M & N are homogeneous expression in x and
y and of same degree then express equation as
Put
Simplify so as to get variable separable diff. eqn.
Solve it & put
Homogeneous Differential Equation
N
M
dx
dy
dx
dv
xv
dx
dy
vxy &
x
y
v
Differential Equations
22Prof. Yogiraj Mahajan

23. y
dx
dy
yx 2
2
yx
dx
dy
xy
yx
yx
dx
dy
23
34
0
3
3
22
22
yx
yx
dx
dy
0
332
dyyxydxx
dxyxydxxdy
22
0
33
4
2
x
yx
dx
dy
y
0secsectan
22
dy
x
y
xdx
x
y
y
x
y
x
23Prof. Yogiraj Mahajan

24. If M & N are functions of x & y and
Solution is given by
Exact Differential Equation
x
N
y
M
CNdyMdx
xfrom
freeterm
Minconst
ykeep
.
Differential Equations
24Prof. Yogiraj Mahajan

25. 0cossecsincostancos
2
dyyxyxdxyxyx
0sectantan2
222
dyyyxxdxyyxy
0sin22
22
dyyxyxdxyxy
x
dx
dy
yxy
2
secsincos
0sectan3sec
2223
dyyxyxdxy
25Prof. Yogiraj Mahajan

26. 0
222222
ydybyxxdxayx
02362
2222
dyyxyxdxyxyx
0sinlogcos
1
1 dyyxxxdxy
x
y
0324
232
22
dyyxyedxxey
xyxy
011 dy
y
x
edxe
y
x
y
x
26Prof. Yogiraj Mahajan

27. Differential Equations
•The most important application of
integrals is to the solution of differential
equations.
•From a mathematical point of view, a
differential equation is an equation that
describes a relationship among a
function, its independent variable, and
the derivative(s) of the function.
27Prof. Yogiraj Mahajan

28. In Applications
Differential equations arise when
we can relate the rate of change of
some quantity back to the quantity
itself.

29. g
dt
xd
dt
dx
dx
d
dt
dv
2
2
The acceleration of gravity is constant (near the surface
of the earth). So, for falling objects:
the rate of change of velocity is constant
Example (#1)
Since velocity is the rate of change of position, we could
write a second order equation:
g
dt
dv
29Prof. Yogiraj Mahajan

30. 2
kvg
dt
dv
Example (#2)
Here's a better one -- with air resistance, the acceleration of a
falling object is the acceleration of gravity minus the
acceleration due to air resistance, which for some objects is
proportional to the square of the velocity. For such an object we
have the differential equation:
rate of change of velocity is
gravity minus
something proportional to velocity squared
or
2
2
2
dt
dx
kg
dt
xd
30Prof. Yogiraj Mahajan

31. In a different field:
Radioactive substances decompose at a rate
proportional to the amount present.
Suppose y(t) is the amount present at time t.
Example (#3)
rate of change of amount is
proportional to the amount (and decreasing)
yk
dt
dy
31Prof. Yogiraj Mahajan

32. Other problems that yield
the same equation:
In the presence of abundant resources (food and
space), the organisms in a population will reproduce
as fast as they can --- this means that
the rate of increase of the population will be
proportional to the population itself:
Pk
dt
dP
32Prof. Yogiraj Mahajan

33. ..and another
The balance in an interest-paying bank
account increases at a rate (called the interest
rate) that is proportional to the current
balance. So
kB
dt
dB
33Prof. Yogiraj Mahajan

34. More realistic situations for the
last couple of problems
For populations: An ecosystem may have a maximum capacity to
support a certain kind of organism (we're worried about this
very thing for people on the planet!).
In this case, the rate of change of population is proportional both
to the number of organisms present and to the amount of excess
capacity in the environment (overcrowding will cause the
population growth to decrease).
If the carrying capacity of the environment is the constant Pmax ,
then we get the equation:
PPkP
dt
dP
max
34Prof. Yogiraj Mahajan

35. and for the Interest
Problem...
For annuities: Some accounts pay interest but
at the same time the owner intends to
withdraw money at a constant rate (think of a
retired person who has saved and is now living
on the savings).
35Prof. Yogiraj Mahajan

36. Another application:
According to Newton's law of cooling , the
temperature of a hot or cold object will change
at a rate proportional to the difference between
the object's temperature and the ambient
temperature.
If the ambient temperature is kept constant at
A, and the object's temperature is u(t), what is
the differential equation for u(t) ?
36Prof. Yogiraj Mahajan

37. A differential equation of the form
gives geometric information about the graph of y(x).
It tells us:
),( yxf
dx
dy
Geometry of Differential Equations
If the graph of y(x)
goes through the
point (x,y), then the
slope of the graph at
that point is equal to
f(x,y).
),( yxf
dx
dy
37Prof. Yogiraj Mahajan

38. Differential Equations
Thanks
for
Silent leasing