Lecture of deferential equation of book advanced

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1. Slide number 1
Differential Equations
Dr. Noor Badshah
www.mathshub.pk
www.youtube.com/@noorshah

2. Slide number 2

3. Slide number 3
Course Contents:
•Introduction to
Differential Equations
•Definitions
•IVP & BVP
•Differential equations as
Mathematical Models
•First-order Differential
Equations
•Formation of ODEs
•Separable equations
•Homogeneous equations
•Linear Equations and
Bernoulli Equations
•Exact Equations/
Reduction to exact
•Higher-order Differential
Equations
•Reduction of order method
•IVP and BVP
•Homogeneous equation
with constant coefficients
•Method of undetermined
coefficient
•Method of variation of
parameter
•Cauchy-Euler Equations
•Laplace Transforms
•Series Solution
•System of Differential
Equations

4. Slide number 4
What is Differential Equation?
•An equation containing the derivatives of one or
more dependent variables, with respect to one or
more independent variables, is said to be a
differential equation (DE).
•Examples:
1.
??????�
??????�
=�
2
2. �
??????
2
�
????????????
2
+
??????�
????????????
+�=�
??????
3. ���−sin���=0

5. Slide number 5
Classification of DE
•Ordinary Differential Equation
If a differential equation contains only ordinary derivatives of one or
more functions with respect to a single independent variable.
•Partial Differential Equation
An equation involving only partial derivatives of one or more
functions of two or more independent variables.

6. Slide number 6
Ordinary Differential Equations
•Where do ODEs arise?
•Notation and Definitions
•Solution methods for 1
st
order ODEs

7. Slide number 7
Where do ODE’s arise
•All branches of Engineering
•Economics
•Biology and Medicine
•Chemistry, Physics etc
Anytime you wish to find out how
something changes with time (and
sometimes space)

8. Slide number 8
Example – Newton’s Law of Cooling
•This is a model of how the temperature of an
object changes as it loses heat to the surrounding
atmosphere:
Temperature of the object:ObjT Room Temperature:Room
T
Newton’s laws states: “The rate of change in the temperature of an
object is proportional to the difference in temperature between the object
and the room temperature”)(
RoomObj
Obj
TT
dt
dT
−−=
Form
ODEt
RoominitRoomObj
eTTTT
−
−+= )(
Solve
ODE
Where is the initial temperature of the object.init
T

9. Slide number 9
Notation and Definitions
•Order
•Linearity
•Homogeneity
•Initial Value/Boundary value problems

10. Slide number 10
Order
•The order of a differential equation is just
the order of highest derivative used.0
2
2
=+
dt
dy
dt
yd
.
2
nd
order3
3
dt
xd
x
dt
dx
=
3
rd
order

11. Slide number 11
Linearity
•The important issue is how the unknown y appears in
the equation. A linear equation involves the
dependent variable (y) and its derivatives by
themselves. There must be no "unusual" nonlinear
functions of y or its derivatives.
•A linear equation must have constant coefficients, or
coefficients which depend on the independent variable
(t). If y or its derivatives appear in the coefficient the
equation is non-linear.

12. Slide number 12
Linearity - Examples0=+y
dt
dy
is linear0
2
=+x
dt
dx
is non-linear0
2
=+t
dt
dy
is linear0
2
=+t
dt
dy
y
is non-linear

13. Slide number 13
Linearity – Summary
Linear Non-lineary2 dt
dy dt
dy
y dt
dy
t 2






dt
dy yt)sin32(+ yy)32(
2
− 2
y )sin(y
or

14. Slide number 14
Linearity – Special Property
If a linear homogeneous ODE has solutions:)(tfy= )(tgy=
and
then:)()( tgbtfay +=
where a and b are constants,
is also a solution.

15. Slide number 15
Linearity – Special Property
Example:0
2
2
=+y
dt
yd 0sinsinsin
)(sin
2
2
=+−=+ ttt
dt
td 0coscoscos
)(cos
2
2
=+−=+ ttt
dt
td 0cossincossin
cossin
)cos(sin
2
2
=++−−=
++
+
tttt
tt
dt
ttd tysin= tycos=
has solutions and
Checktty cossin+=
Therefore is also a solution:
Check

16. Slide number 16
Homogeniety
•Put all the terms of the equation which involve the dependent
variable on the LHS.
•Homogeneous: If there is nothing left on the RHS the equation is
homogeneous (unforced or free)
•Nonhomogeneous: If there are terms involving t (or constants) -
but not y - left on the RHS the equation is nonhomogeneous (forced)

17. Slide number 17
Example
•1st order
•Linear
•Nonhomogeneous
•Initial value problemg
dt
dv
= 0
)0(vv= w
dx
Md
=
2
2 0)(
0)0(
and
=
=
lM
M
◼2nd order
◼Linear
◼Nonhomogeneous
◼Boundary value
problem

18. Slide number 18
Example
•2nd order
•Nonlinear
•Homogeneous
•Initial value problem
◼2nd order
◼Linear
◼Homogeneous
◼Initial value problem0sin
2
2
2
=+ 

dt
d 0)0( ,0
0
==
dt
d
θ)θ(
 0
2
2
2
=+

dt
d 0)0( ,0
0
==
dt
d
θ)θ(


19. Slide number 19
Solution Methods - Direct Integration
•This method works for equations where
the RHS does not depend on the
unknown:
•The general form is:)(tf
dt
dy
= )(
2
2
tf
dt
yd
=  )(tf
dt
yd
n
n
=

20. Slide number 20
Direct Integration
•y is called the unknown or dependent variable;
•t is called the independent variable;
•“solving” means finding a formula for y as a function of t;
•Mostly we use t for time as the independent variable but in some
cases we use x for distance.

21. Slide number 21
Direct Integration – Example
Find the velocity of a car that is
accelerating from rest at 3 ms
-2
: 3==a
dt
dv ctv+= 3 tv
ccv
3
00300)0(
=
=+==
If the car was initially at rest we
have the condition:

22. Slide number 22
Solution of ODE:
•A function ??????(�) is said be solution of an ODE, if it satisfies the given
ODE.

23. Slide number 23
Types of Solutions of DEs:
•General Solution
�
′
=??????� has solution of the form �=�
0�
????????????
, is the general solution.
•Particular Solution
For the same example, if we apply condition �0=1, the solution
becomes �=�
????????????
is the particular solution.
•Singular Solution
For a differential equation �
′
=??????�, the solution is �=
1
4
�
2
+�

2
is
the general solution. The equation has solution �=0 which can not
be obtained from the general solution, so this solution is called
singular solution.

24. Slide number 24
Exercise 2.1

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32. Slide number 32
Differential Equations as Mathematical Models

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