Differential Equation

Fist Order Ordinary Differcntial Equations




1.1 An Introduction to Differential Equations
Definition
Differential Equations
A differential equation is an equation that contains a derivative (or
derivatives) of an unknown function.
A differential equation is ordinary if all derivatives are with respect to a
single independent variable, such as

or or y, y,..y,
A differential
equation is partial if there derivatives with respect to two
more independent variables, such as
are
or|
du ou o'u



Note that the notation for ordinary derivative i sd , for partial derivative, we


use.Ourdiscussion throughout the book will be concentrated on ordinary
Ox
differential equations (ODE), or simply known as differential equation.
Besides classify the differential equation according to its derivatives,
another classification of differential equations is in terms of their order.

Definition
The Order of a Differential Equation
The order of a differential equation is the order of the highest derivative in the
equation.

Some examples of the ordinary differential equations (ODE) are
given below.

= -kx, where k is a constant. (first order ODE)
y-2y+5y= cos 2x (second order ODE)

++8y=2e"
dx3 dx
(third order ODE)
2
3-2 ax (tirst order ODE)

Meanwhile, in many engineering application, deal with
differential equation (PDE), such as
we
partial

, First Order Ordinary Differential Equations




Heat equation: 0 , where c is a constant. (second order PDE)
c
Wave equation: 0 , where c is a constant. (second order PDE)

Laplace'sequation:+ a'u =0 (second order PDE)


2-2x +4y first order PDE)


Example 1.1.. ***************************************************e****** *e******
Determine whether the following equation is an ordinary or partial differential
equation. For each differential equation, write down the order of differential
equation.

(a)2 3=0 s)+(cosz)ptr)=0 x-*
(d)+y=2 +Dy9-n-
Solution:
(a) Second order ordinary differential equation
(b) Second order ordinary differential equation
(c) First order partial differential equation
(d) Not a differential equation
(e) Fourth order ordinary differential equation
. Iry Question 1, Exercise 1A
1.1.1 The Formation of Ordinary DifferentialEquations
In many engineering applications, we employ certain physical laws to
model the phenomena into an ordinary differential equation (ODE). Besides,
ODE can also be formed by eliminating the constant(s) in the given
expression. Basically, we need to, carry out n times differentiation to eliminate
all the constant in an expression with n constants.


Example12. *eco** **e* *****

Obtain an ordinary diferential equation from the expression y=2e2

Solution:
Step 1:One arbitrary constant, that is 4, hence get y.

y-4 4 ()
Step2: Fom a differential equation.
From y=2er 4 rearrange it by writing the constant d as the

subject offomula, we get A=*'y-2*). Substitute in cquation (),
weobtain = - 4 ¢ _ 2 ( y - 2 e * ) e : . .

, First Order Ordinary Differential Equations




For expression with one constant, we obtain first order ordinary
Note
differential equation.
...Try Question 2(a) (b). Exercise 1A
-




Example 1.3. the expression
Obtain an ordinary differential equation from
(b) = Asin 2x + Bcos 2x
(a) y= Ae + Bx
by eliminating the constants.
Solution:
hence get y' and y.
(a) Step l: Two constants (A and B),
y= Ae + B (i)
Ae A=ey'--- (i)
Step 2: Fom a differential equation.
From (i), B =y- A e (i)
Substitute (i) and (ii) in the given equation,
y=e"ye" +(y-Ae)x
where A e"y"
=y+3-Axe"
=y+3y-e"y'xe
-y = 0


(b) Step]: Two constants (A and B), hence get y' and y'".
y 2Acos2x-2Bsin 2x
'= -4Asin 2x-4Bcos 2x
Step 2: Form a differential equation.
=-4(Asin 2x + Bcos2x)
-4y
Note: For expression with two constants, we obtain second order ordinary
differential equation.
T r y Question 2(c) - (e), Exercise 1A



1.1.2 The Solution of Ordinary Differential Equations
Definition
The Solution of an Ordinary Differential Equation
A function is solution of
a
ordinary differential equation if it satisties
an
differential equation, which makes the equation true (or left-hand side (LH
equal to right-hand side (RHS).

Such a function with one or more
arbitrary constant(s)
solution of the ordinary differential equation.
is known as gener
When numerical value(s) 13
given to the
arbitrary constant(s) in the general solution, it results in
a
particular solution of the ordinary differential
equation.