The document provides an overview of ordinary differential equations (ODEs), fundamental in modeling dynamic systems. It explains the basic types, including first-order, second-order, linear, and nonlinear ODEs. Key methods for solving ODEs, such as separation of variables, integrating factors, and...
LESSON 1
Differential Equations and Their Solutions
1.1 Introduction
The subject of differential equations constitutes a very important and useful branch of
modern mathematics. In this lesson we shall consider some definition of ordinary
differential equations.
1.2 Objectives of the lesson
By the end of this lesson you should be able to.
define a differential equation
define the order and the degree of a differential equation.
form the differential equation and solve simple differential equations.
1.3 Definition of ordinary differential equation
dy d 2 y
An equation containing any derivatives such as , , is called an ordinary
dx dx 2
differential equation.
Examples of differential equations
dy
1. = 5x
dx
dy
2. + 3x + 5 = 0
dx
d2y dy
3. 2
+8 + 9y = 0
dx dx
d2y
4. + y = sin x
dx 2
are ordinary differential equations.
1.4 Definition of Partial Differential Equations
A differential equation involving partial derivatives of a dependent variable, with respect
to more than one independent variable, is called a partial differential equation.
For example,
∂
(2 x 2 + 5 y 2 + 3xy 2 ) = 0
∂x
∂
(sin x + 5 y cos x + x 2 y 3 ) = 0
∂x
∂2 3
( x + 3 x 2 y + y 2 ) = 0 are called partial differential equations. This means when you
∂y 2
differentiate a term with respect to x, y is treated as constant similarly x is treated as a
constant when we differentiate a term with respect to y.
,ELITE TUTORING
∂ ∂ 3
Then (3 y 2 x 3 ) = 3 y 2 x = 9x2 y 2
∂x ∂x
∂ ∂
(7 y 2 sin x) = 7 sin x y 2 = 14 y sin x
∂y ∂x
d2y d3y dny
1.5 Meaning of , ………
dx 2 dx 3 dx n
d2y dy
2
means that we differentiate again with respect to x.
dx dx
d2y d dy
=
dx 2
dx dx
d3y d d 2 y
=
dx 3 dx dx 2
dny
similarly is defined.
dx n
1.6 The order of a differential equation
The order of a differential equation is the order of the highest differential coefficient
present in the equation.
For example the equation,
d3y d2y dy
7 3
+ 8 2
+9 + 8y = ex
dx dx dx
dy d 2 y d 3 y d3y
contains, , , the highest differential coefficient is . Hence the order of
dx dx 2 dx 3 dx 3
the equation is three.
10
d 2 y dy
Consider 3 2 + + 8y = 0
dx dx
d2y
The highest differential coefficient present in this equation is . Hence the order of
dx 2
the equation is two.
1.7 The degree of a differential equation
The degree of a differential equation is the degree (or power) of the highest differential
coefficient present, when the differential coefficient are free from radicals and fractional
exponents.
,ELITE TUTORING
For example consider the equation,
10 7
d3y d2y dy
3
+ 2 + 3 + 8 y = 0
dx dx dx
d3y d3y
The highest order present in this equation is . The degree of this is one. Hence
dx 3 dx 3
the degree of the differential equation is just one.
Example 1
Consider,
5 11 20
d 3 y d 2 y dy
3 + 2 + + y = x
dx dx dx
d3y
Here the highest differential coefficient is or the order of the equation is 3. The
dx 3
degree of this highest differential coefficient is five and hence the degree of this equation
is five.
Example 2
3
d 2 y 2 dy
Consider, 2 + 5 + 3 y = 0
dx dx
d2y
The highest differential coefficient present here is . First it should be free from
dx 2
radicals and fractional exponents.
3
d 2 y 2 dy
2 = −5 − 3 y
dx dx
3 2
d 2 y dy
2 = − 5 − 3 y
dx dx
Then the degree of the equation is three.
Consider,
2
dy
3 + = 2x
dx
2
dy
Removing the radical it is written as 3 + = 4 x 2 .
dx
Hence the degree of the equation is two.
1.8 The solution of a differential equation
, ELITE TUTORING
The solution of a differential equation is an equation between x and y, which will satisfy
the differential equation.
Example 3
2 d2y dy
a) Show that y = x is a solution of the equation, 3 2 + 5 = 10x + 6
dx dx
b) Check whether y = x2 + c where c is any constant is also a solution of the equation.
Solution
Let y = x2 (1)
dy
= 2x (2)
dx
d2y
=2 (3)
dx 2
substituting (1), (2) and (3) in the given equation
3(2) + 5 (2x) = 6 + 10x thus the equation is satisfied
1.9 General and particular solution of a differential equation.
dy
Consider =2 (1)
dx
Then dy = 2dx
Integrating, ∫ dy = ∫ 2dx
y = 2x + c where c is any arbitrary constant
we say that y = 2x + c is the general solution of the equation (1)
Geometrically, y = 2x + c is a family of straight lines parallel to y = 2x
The solution y = 2x + c, represents a family of straight lines. This family y = 2x + c is
dy
called the general solution of the equation = 2.
dx
When c takes a particular value (say c = 3) we get a particular member of the family, y =
2x + 3. It is a particular solution.
A particular solution is one where a value is given to c.
dy
In the above example, the order of the equation = 2, is one. Hence the general
dx
solution of this equation contains one arbitrary constant c.
In general, the solution of an nth order differential equation will have n arbitrary
constants. For example the general solution of a second order equation will have two
arbitrary constants. When we give particular values for these constants we get particular
solutions of the equation.
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