3. I. Definitions
Learning Objective
At the end of the section, you should be able to
define a differential equation and classify
differential equations by type, order and linearity.
5. I. Definitions
What is a Differential Equation
A differential equation (DE) is an equation containing the
derivatives of one or more dependent variables with
respect to one or more independent variables.
8. I. Definitions
Classification by Type
Two types of Differential equations (DE) exist:
• ORDINARY DIFFERENTIAL EQUATION (ODE).
An equation containing only ordinary derivatives of one
or more dependent variables with respect to a SINGLE
independent variable is said to be an Ordinary
Differential Equation (ODE).
9. I. Definitions
Examples of ODE
x
e
y
dx
dy
5
)
1
0
6
)
2 2
2
y
dx
dy
dx
y
d
y
x
dt
dy
dt
dx
2
)
3
10. I. Definitions
• PARTIAL DIFFERENTIAL EQUATIONS (PDE).
An equation containing partial derivatives of one or more
dependent variables with respect to TWO or more
independent variables is said to be a Partial Differential
Equation (PDE).
11. I. Definitions
Examples of PDE
0
)
1 2
2
2
2
y
u
x
u
t
u
t
u
x
u
2
)
2 2
2
2
2
x
v
y
u
)
3
12. I. Definitions
Classification by Order
The order of a differential equation (ODE or PDE)
is the order of the highest derivative in the
equation.
13. I. Definitions
Examples of Orders
x
e
y
dx
dy
3
5
0
6
2
2
y
dx
dy
dx
y
d
x
e
y
dx
dy
dx
y
d
4
5
3
2
2
is of order 1 (or first-order)
is of order 2
is of order 2
15. I. Definitions
Classification by Linearity
x
g
y
x
a
y
x
a
...
y
x
a
y
x
a n
n
n
n
0
1
1
1
The general form for an nth-order ODE is:
x
g
y
x
c
y
x
b
y
x
a
The general form for an 2nd-order ODE is:
16. I. Definitions
Examples for linear ODEs
0
4
)
1
dy
x
dx
x
y
0
2
)
2
y
y
y
x
e
y
dx
dy
x
dx
y
d
5
)
3 3
3
x
y
y
x
4
17. I. Definitions
Examples for non-linear ODEs
x
e
y
y
y
-
2
1
)
1
0
sin
)
2 2
2
y
dx
y
d
0
)
3 2
4
4
y
dx
y
d
19. I. Definitions
Solution:
ODE Order Linearity
0
cos
dx
x
xy
dy
0
6
2
2
dt
dQ
dt
Q
d
0
2
2
xy
y
y
y
x
y
0
y
y
x
ey
Linear
Linear
Non-linear
Non-linear
1
2
3
2
20. I. Definitions
Solution:
ODE Order Linearity
sin
0
1
2
xdy
dx
y
2
2
2
1
dx
dy
dx
y
d
Linear
Non-linear
Non-linear
1
1
2
2
sin
21. I. Definitions
Exercise-I:
For each of the following ODEs, determine the order and
state whether it is linear or non-linear:
t
y
dt
dy
t
dt
y
d
t sin
2
2
2
2
t
e
y
dt
dy
t
dt
y
d
y
2
2
2
)
1
(
1
2
2
3
3
4
4
y
dt
dy
dt
y
d
dt
y
d
dt
y
d
0
2
ty
dt
dy
t
y
t
dt
y
d
sin
)
sin(
2
2
3
2
3
3
)
(cos t
y
t
dt
dy
t
dt
y
d
t
ty
dt
dy
t tan
1
2
22. II. Classification of Solutions
Learning Objective
At the end of this section, you should be able to
• verify the solutions to a given ODE
• identify the different types of solutions of an
ODE.
• Define IVP, BVP
23. Definition:
A solution of a DE is a function that satisfies the DE
identically for all in an interval , where is the
independent variable.
y
x I x
II. Classification of Solutions
24. Example
x
y ln
)
,
0
(
I
is a solution of the DE: 0
'
"
y
xy
x
y ln
x
y
1
'
2
1
"
x
y
x
x
x
y
xy
1
)
1
(
'
'
' 2
Indeed,
II. Classification of Solutions
0
1
1
x
x
25. Definition:
A solution in which the dependent variable is expressed
solely in terms of the independent variable and constants
is said to be an explicit solution.
II. Classification of Solutions
26. Definition:
A solution in which the dependent and the independent
variables are mixed in an equation is said to be an implicit
solution.
II. Classification of Solutions
27. Examples:
x
y ln
is an explicit solution of the DE: 0
'
"
y
xy
9
2
2
y
x
9
2
2
y
x
is an implicit solution of the DE: 0
'
x
yy
Indeed:
Implicit differentiation: 0
'
2
2
yy
x
0
'
yy
x
1)
2)
II. Classification of Solutions
28. General or Particular solution
Example:
Consider the ODE: 0
'
y
y
x
e
y is a solution (particular)
x
ce
y (where c is a constant) is a solution (general)
II. Classification of Solutions
x
e
y 2
is also a solution (particular)
29. General or Particular solution
• A solution of a DE that is free of arbitrary parameters is
called a particular solution.
II. Classification of Solutions
Definitions:
• A solution of a DE representing all possible solutions is
called a general solution.
30. Example
II. Classification of Solutions
x
ce
y is a 1-parameter family of solutions of the DE
0
'
y
y
x
x
de
ce
y
is a 2-parameter family of solutions of the DE
0
"
y
y
31. Example:
Verify that the indicated function is an explicit solution of
the given DE :
II. Classification of Solutions
32. Example:
1)
II. Classification of Solutions
2
;
0
2
x
e
y
y
y
2
2
1
'
x
e
y
2
2
)
2
1
(
2
'
2
x
x
e
e
y
y
0
2
2
x
x
e
e
33. Example:
2)
II. Classification of Solutions
t
e
y
;
y
dt
dy 20
5
6
5
6
24
20
t
e
dt
dy
y 20
)
5
6
(
20
'
t
e 20
24
t
t
e
e
y
dt
dy 20
20
5
6
5
6
20
24
20
t
t
e
e 20
20
24
24
24
24
34. 3)
II. Classification of Solutions
x
cos
e
y
;
y
y
y x
2
0
13
6 3
x
e
x
e
y x
x
2
sin
2
2
cos
3
' 3
3
x
e
y x
2
sin
2
3 3
x
e
x
e
y
y x
x
2
cos
4
2
sin
6
'
3
" 3
3
y
x
e
x
e
y x
x
4
2
sin
6
)
2
sin
2
3
(
3 3
3
y
y
y 13
6 y
x
e
y
x
e
y x
x
13
)
2
sin
2
3
(
6
2
sin
12
5 3
3
Example:
x
e
y x
2
sin
12
5 3
0
13
2
sin
12
18
2
sin
12
5 3
3
y
x
e
y
x
e
y x
x
35. 4)
II. Classification of Solutions
x
tan
x
sec
ln
x
cos
y
;
x
tan
y
y
x
x
x
x
x
y sec
cos
tan
sec
ln
sin
'
1
tan
sec
ln
sin
x
x
x
x
x
x
x
x
y sec
sin
tan
sec
ln
cos
"
x
x
x
x tan
tan
sec
ln
cos
x
x
x
x
x
x
x
y
y tan
sec
ln
cos
tan
tan
sec
ln
cos
x
tan
Example:
36. 5)
II. Classification of Solutions
t
t
e
c
e
c
P
P
P
P
1
1
1
;
1
t
t
e
c
e
c
P
1
1
1
2
1
1
1
1
1
1
1
'
t
t
t
t
t
e
c
e
c
e
c
e
c
e
c
P
2
1
1
2
1
2
2
1
2
2
1
1
1
1 t
t
t
t
t
t
e
c
e
c
e
c
e
c
e
c
e
c
t
t
t
t
e
c
e
c
e
c
e
c
P
P
1
1
1
1
1
1
1
1
t
t
t
t
t
e
c
e
c
e
c
e
c
e
c
1
1
1
1
1
1
1
1
'
1
2
1
1
P
e
c
e
c
t
t
Example:
37. 6)
II. Classification of Solutions
x
x
xe
c
e
c
y
;
y
dx
dy
dx
y
d 2
2
2
1
2
2
0
4
4
x
x
x
xe
e
c
e
c
y 2
2
2
2
1 2
2
'
x
x
x
xe
c
e
c
e
c 2
2
2
2
2
1 2
2
x
x
x
x
e
c
y
e
c
xe
c
e
c 2
2
2
2
2
2
2
1 2
2
x
e
c
y
y 2
2
2
'
2
"
y
y
y 4
'
4
"
y
y
e
c
y x
4
'
4
2
'
2 2
2 y
y
e
c x
4
'
2
2 2
2
0
4
2
2
2 2
2
2
2
y
e
c
y
e
c x
x
Example:
38. Exercise-II:
Verify if the indicated functions are explicit solutions of the
given DE :
II. Classification of Solutions
t
t
t
t
t
y
t
t
y
y
t
t
t
y
t
t
y
t
y
ty
y
t
t
t
y
t
t
y
t
y
ty
y
t
t
e
t
y
t
t
y
t
y
y
y
t
t
y
t
y
y
t
t
sin
cos
ln
)
(cos
)
(
;
2
0
,
sec
'
'
)
5
ln
)
(
,
)
(
;
0
,
0
4
'
5
'
'
)
4
)
(
,
)
(
;
0
,
0
'
3
'
'
2
)
3
3
)
(
,
3
)
(
,
3
4
)
2
3
,
)
1
2
2
2
1
2
1
2
2
1
1
2
2
1
)
3
(
)
4
(
2
2
39. Definition
A DE with initial conditions on the unknown function and its
derivatives, all given at the same value of the independent
variable, is called an initial-value problem, IVP.
0
x
II. Classification of Solutions
41. Definition
A DE with initial conditions on the unknown function and its
derivatives, all given at different values (e.g. at and )
of the independent variable, is called a boundary-value
problem, BVP.
0
x 1
x
II. Classification of Solutions
42. Examples
2
2
,
1
;
2
)
1
y
y
e
y
y x
1
1
,
1
0
;
2
)
2
y
y
e
y
y x
II. Classification of Solutions
43. Examples
Find the solution of the IVP or BVP if the general solution is the
given one:
,
2
3
;
0
)
1
y
y
y x
e
c
x
y
1
3
1
3
e
c
y 2
3
y
2
3
1
e
c
3
1 2e
c
x
x
e
e
e
x
y
3
3
2
2
solution of the IVP:
II. Classification of Solutions
45. solution of the BVP:
6
2
cos
6
2
sin
6
2
1
c
c
y
2
2
3 2
1 c
c
1
6
y 1
2
3 2
1
c
c
2
3 1
1
c
c
1
3
2
1
c
1
3
2
2
c
x
x
y 2
cos
2
sin
1
3
2
Examples
II. Classification of Solutions
46. x
c
x
c
x
y 2
cos
2
sin 2
1
0
cos
0
sin
0 2
1 c
c
y
2
c
1
0
y 1
2
c
,
2
2
,
1
0
;
0
4
)
3
y
y
y
y
cos
sin
2
2
1 c
c
y
2
c
2
2
y 2
2
c
2
2
c
1
2
c
IMPOSSIBLE NO SOLUTION
Examples
II. Classification of Solutions
47. Exercise-III
1) Determine and so that
will satisfy the conditions :
0
8
y 2
8
y
1
c 2
c 1
2
cos
2
sin 2
1
x
c
x
c
x
y
2) Determine and so that
will satisfy the conditions :
x
e
c
e
c
x
y x
x
sin
2
2
2
1
1
c 2
c
0
0
y 1
0
y
II. Classification of Solutions
