Ordinary differential equations notes

MATH 322: ORDINARY DIFFERENTIAL EQUATIONS I

COURSE CONTENT
 First order equations and applications.
 Second order equations. 1.0 INTRODUCTION TO DIFFERENTIAL EQUATIONS
 Homogenous equations with constant A differential equation is an equation that involves one or more
derivatives e.g.
coefficients.
(1)
 Equations with variable coefficients.
 Non-homogeneous equations.
 Undeterminedcoefficients.
 Variation of parameters. (2)
 Inverse differential Operators.

REFERENCES (3)
1.1 Types of differential equations
1. C. Ray Wyle,Alfred L. R. And Brouson,(2002) Differential There are two types of differential equations namely
Equations, Wisley
2. Erwin Kreyszig(1978) Advanced Engineering Mathematics, i. Ordinary Differential equations (ODE)
These are differential equations in which there is only one
John Wiley and Sons independent variable.
3. James Stewart(2003) Calculus, Early Transcendentals, They are ordinary derivatives of one or more dependent variable
Thomson, 5th Edition, International Student Edition w.r.t. a single independent variable e.g.
4. Schaum’s Series(1974)Theory and Problems of Differential
Equations
5. Wilcox and Rainville(1991) Elementary Differential y – Dependent variable and x – independent variable
Equation (1) and (2) above are ordinary differential equations
Equations, Lagas vegas


1

, ii. Partial differential equation (PDE) Example 1
These are equations in which dependent variable depend on more State the order and degree of the differential equations below.
than one independent variable. The derivatives involved are partial


(4)
v – Dependent variable and x, y, z – independent variables.
Equation (3) and (4) above are partial differential equations.

1.2:DEFINITIONS OF TERMS
a) Order of a differential equation
The order of a differential equation is the order of the highest
derivative appearing in the equation e.g.

Order 2 Solution

Order 3 Equation Order Degree
4 1
b) degree of a differential equation
The degree of a differential equation is the power of the highest 2 4
ordered derivative which occurs in the equation e.g.

3 1
Degree 1

7 4




2

, iii.
1.3 CLASSIFICATION OF ORDINARY DIFFERNTIAL
EQUATIONS
ORIGIN OF DIFFERENTIAL EQUATIONS
Ordinary differential equations are classified into two groups
Differential equations occur in connection to numerous problems
a) Linear differential equations
that one encounters in various branches of science and engineering.
A differential equation is said to be linear if the dependent variable
i. e. differential equations may arise from
and its derivative occur only in the first degree and are not
a) Geometric problems
multiplied together. The general form of a linear differential
b) Physical problems
equation of order n in the dependent variable y is expressed in the
c) Primitive.
form
Examples
i. A falling stone from the top of a building if x is time and y is
displacement then
Where
Examples of linear equations are
Hence we can write a model


By integrating we get
The Equation above is a linear equation with constant coefficient By integrating once more we get the displacement as
b) Non-linear differential equations
If the O.D.E. is not linear then it is non-linear.

Example 2.
Identify if the equations below are linear or non - linear
Where is the initial velocity and is the displacement when t =
0
i. ii. Determination of motion of a projectile. e.g. rocket
iii. Determination of the charge or current in an electric circuit

ii. If charge and current at time t are related by

3

, Where E is voltage, L inductance and R resistor Where k = B+C and therefore there are only 2 essential constants
The general solution is A primitive involving n-essential arbitrary constant gives rise to
differential equations of order n free of arbitrary constant.
We obtain equation by eliminating the n constants by differentiating
n times.

iv. Determining of the vibration of a wire or a membrane Example 3
v. Heat condition in a rod or in a slab Obtain the differential equation associated with
vi. Population growth in human, animal and bacteria e.t.c
Solution
The mathematical formulation of such problems gives rise to
differential equations ........................................................ (i)
........................................................ (ii)
Primitive
A primitive is a relation between variables which involve n – ........................................................ (iii)
essential arbitrary constant e.g. Subtracting (i) from (ii) we get

Subtracting (ii) from (iii) we get


NB: Arbitrary constant are said to be essential if they cannot be
replaced by smaller numbers of constants. for example given
Example 4
A, B and C are not all essential constants because they can be Show that a function defined by where and
reduced to 2 constants
are arbitrary constants is associated with a differential equation of
the form
4