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Lecture 2: Engineering Mathematics 2A: Chapter 1: Introduction To Differential Equations

This document provides an overview of differential equations including: 1) Classifying differential equations as ordinary or partial, and by order and linearity 2) Defining solutions of differential equations 3) Introducing four methods for solving first-order differential equations including separable equations

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75 views23 pages

Lecture 2: Engineering Mathematics 2A: Chapter 1: Introduction To Differential Equations

This document provides an overview of differential equations including: 1) Classifying differential equations as ordinary or partial, and by order and linearity 2) Defining solutions of differential equations 3) Introducing four methods for solving first-order differential equations including separable equations

Uploaded by

terrygoh
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PPT, PDF, TXT or read online on Scribd
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Lecture 2 : Engineering

Mathematics 2A
Chapter 1 : Introduction to Differential Equations
1.1 : The classification of differential equations
1.1 : Verification of a solution
1.1 : Using different symbols
1.3 : Differential Equations as Mathematical Models

Chapter 2 : First-Order Differential Equations


2.1.1 : Direction fields

Methods Solving Differential Equations


2.2 : Separable Variables
Introduction to Differential Equations
 p2 : Definition 1.1.1 Differential Equation

An equation containing derivatives of

one or more dependent variables,

with respect to

one or more independent variables,

is said to be a differential equation (DE).


Introduction to Differential Equations
 p2 : Definition 1.1.1 Differential Equation
Classification of Differential Equations
Classify Simplified Definition Reference
By
1. Ordinary DE ( ODE) p2
Type If DE contains only ordinary derivatives.
example Last para
of DE 2. Partial DE ( PDE )
If DE contains partial derivatives. example
Order
of DE
1. Derivatives notation

2. Order of a DE is the order of the highest derivatives
p3

in the equation . example


Linearity Linear nth-order ODE : p4
of DE dny d n1 y dy
an  x  n  an1  x  n1    a1  x   a0  x  y  g  x 
dx dx dx
properties
Classification by Type
 p2 : Examples of O.D.E :
dy
 5 y  ex
dx

d y dy
2

  6y 0
dx dx
2

dx dy
  2x  y
dt dt

RETURN
Classification by Type
 p3 : Examples of P.D.E :
u u
2 2

 2 0
x 2
y

u u
2 2
u
 2 2
x t
2
t

u v

y x

RETURN
Classification by Type
 p3 : Notation for Ordinary Derivatives.
Throughout this text order of Leibniz Prime
ordinary derivatives derivative Notation Notation
1st dy y
will be written by using dx
either the 2nd d2y y
dx 2
Leibniz notation or the 3nd d3y y
prime notation. dx 3

dy
 5 y  e1x
example can be written as 
y  5y e x

dx
dexample
2
y dy 2 y  y  6 y  0
  6y 0 can be written as
dx dx
2
Classification by Type
 p3 : Notation for Ordinary Derivatives.
Actually, the prime notation is used to denote only the
first three derivatives ; the fourth derivative is written
y  4  instead of y.

th d n
y
In general, the n derivative of y is written as n or y  n  .
dx

RETURN
Classification by Order
 p3 : Classification by order
3
d y
2
 dy 
 5   4 y  e x

dx  dx 
2

RETURN
Classification by Linearity
 p4 : Classification by Linearity

An nth-order ODE is linear when


dny d n1 y dy
an  x  n  an1  x  n1    a1  x   a0  x  y  g  x 
dx dx dx
Classification by Linearity
 p4 : Classification by Linearity
Any ODE must satisfy the 2 properties in order to be a
linear ODE otherwise it is a nonlinear ODE.
Two properties of a linear ODE are as follows :
 The dependent variable y and all its derivatives
y, y, , y  n  are of the first degree, that is, the
power of each term involving y is 1. example

 The coefficients a0 , a1 , , an of y, y, , y  n  depends


at most on the independent variable x. example
RETURN
Classification by Linearity
 p4 : Example of linear / non-linear O.D.E :
d4y
 y 2
0
dx 4

d4y
 x 2
y 0
dx 4

2
d y 
4

 4   y 2
0
 dx 

d y
2

 sin y  0
dx 2
Classification by Linearity
 p4 : Example of linear / non-linear O.D.E :
d4y
y 4  cos x  0
dx
d4y
 y cos x  0
dx 4

d y
3
dy
x  5y 0
dx 3
dx

RETURN
Solution of an ODE.
 What does a solution(s) of an ODE mean ?
Suppose we have an equation y  f  x, y  ,
if we can differentiate this equation n th times
then we obtained an nth-order ODE.
We say that
The equation y = f(x,y) is
the solution of the nth-order ODE.
To solve an ODE means
to find the solution of the ODE.
Solution of an ODE.
 p2 : example

y e 0. 1 x 2
(1)

differentiate (1) we get a 1st -order ODE :


dy
 0.2 xy (2)
dx
differentiate (2) we get a 2nd -order ODE :

d2y  dy 
 0.2 y  x  (3)
dx  dx 
2
Solution of an ODE.
 Discussions :
dy
_______________ is the solution of  0.2 xy
dx

d2y  dy 
_______________ is the solution of 2  0.2 y  x 
dx  dx 
Solving an ODE
st dy
 Now suppose we given the 1 -order ODE,  0.2 xy
dx
How do we verify that the solution of this ODE is
y e 0.1 x 2
and not y  e ?
x2

solution :
Solving of an ODE.
 To obtain the 1st-order ODE, we differentiate the
solution.
 Theoretically, to obtain the solution we can
integrate the 1st-order ODE.
Unfortunately, integration alone is not enough to
solve any 1st-order ODE. There are 4 methods that
we will study in this module to solve 1st-order
ODE.
Solving 1 Order st

Differential Equation
Method 1 : Separable Equation

Method 2 : Integrating Factor

Method 3 : Exact Equation

Method 4 : Euler’s Method ( Numerical Method )


Method 1: Separable Equation
 p45 : Definition 2.2.1 Separable Equation
If a 1st-order ODE can simplified to the form

dy
 g  x  h y 
dx
is said to be separable or to have separable
variables.
Method 1: Separable Equation
 Strategy to solve a separable equation
1. Identify the ODE is a separable equation.

2. Regroup the separable equation to the form


  dy    dx (1)

3. Integrate equation (1)


  dy     dx
Method 1: Separable Equation
 p46 : Example 1
Solve 1  x  dy  y dx  0
Method 1: Separable Equation
 p47 : Example 4
dy
Solve  e  y  cos x
2y
 e y sin 2 x , y  0   0.
dx

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