1.0 INTRODUCTION
Solving differential equation is one of the major problems of
numerical analysis. This is because wide variety of applications
lead to differential equations, and so few can be solved analytically.
The classical initial value problem is to find a function which𝑦(𝑥)
satisfies the first order differential equation and takes𝑦 = 𝑓 𝑥, 𝑦( )
the initial value . A broad variety of methods have been𝑦 𝑥0( ) = 𝑦0
devised for the approximate solution of this problem, most of which
have been generalized for treating higher order problems as well.
Differential equations have long fascinated mathematicians both
because of their intrinsic interest and because of their application in
such disciplines as physics, chemistry, engineering, astronomy,
economics and psychology. The fact that differential equations
arise in such variety of fields is hardly surprising when we recall hat
the derivative of a function can be interpreted as the
𝑑𝑦
𝑑𝑥 𝑦 = 𝑓 𝑥( )
rate of change of with respect to . Thus, descriptions of natural𝑦 𝑥
phenomena that involve change are often given in terms of
differential equations.
Many of the laws in physics, chemistry, engineering, biology and
economics are based on empirical observations that describe
changes in the states of systems. Mathematical models are often
expressed in terms of not only certain system parameters but also
their derivatives. Such mathematical models, which use differential
calculus to express relationship between variables, are known as
differential equations.
Examples of differential equations are many. Few of them are listed
below to illustrate the types of differential equations that occur in
science and engineering.
i. One of the most pervasive phenomena in physical universe is
motion and the most famous differential equation of all time
is Newton’s second law of motion. i.e. which𝑓 = 𝑑 𝑚𝑣( )
𝑑𝑡
states that the force acting on an object of mass that is𝐹 𝑚
moving with velocity is equal to the time derivative of the𝑉
momentum of the object. In this equation and are𝐹 𝑉
vectors quantities i.e. quantities which have both magnitude
and direction.
ii. Law of cooling states that the rate of loss of heat from a
liquid is proportional to the difference of temperature
between the liquid and the surroundings. This can be stated
in mathematical form as where is
𝑑𝑇 𝑡( )
𝑑𝑡 = 𝐾 𝑇𝑠 − 𝑇 𝑡( )( ) 𝑇𝑠
the temperature of surroundings, is the temperature of𝑇 𝑡( )
liquid at time t and k is the constant of proportionality.

iii. Radioactive decay
The radioactive decay of an element is given by
where is the mass is the time and is
𝑑𝑚
𝑑𝑡 − 𝑘𝑛 = 0 𝑚 𝑡 𝑘
the constant of rate of decay. Note that all these examples
contain the rate of change of variable expressed as function
of parameters. Although most of the differential equations
may be solved analytically in their simplest form, analytical
techniques fail when the models are modified to take into
account the effect of other conditions of real life situations.
In all such cases, numerical approximation of the solution
may be considered as a possible approach [1, 3, 5].
1.1 Origin of Differential Equations
a. Geometric problems
b. Physical problems
c. Primitive
Geometric Problem
A curve is defined by the condition that its point , its slope𝑥, 𝑦( ) 𝑑𝑦/𝑑𝑥
is equal to twice the sum of the coordinates of the point. We want to
expressed the condition by means of a differential equation
representing the condition is 𝑑𝑦
𝑑𝑥 = 2 (𝑥 + 𝑦) [2, 3 ]