Ordinary Differential Equations

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UNIT 1. ORDINARY DIFFERENTIAL EQUATIONS (ODE) OF FIRST-ORDER
AND FIRST DEGREE

1.1 Introduction

Differential Equations

Many physical laws and relations, rules of nature can be expressed mathematically in the
form of differential equations. Indeed, many engineering problems appear as differential equations.
Thus it is natural for an engineer to study differential equations and most important methods for
solving them and modeling also. Modeling is a crucial general process in engineering, physics,
computer science, biology, medicine, environmental science, chemistry, economics, and other fields
that translates a physical situation or some other observations into a “mathematical model.”
Numerous examples from engineering (e.g., mixing problem), physics (e.g., Newton’s law of
cooling), biology (e.g., Gompertz model), chemistry (e.g., radiocarbon dating), environmental
science (e.g., population control), etc.

Definition: Differential Equation
An equation consisting derivatives or differential coefficients of dependent variables with
respect to one or more independent variables is called as differential equation.
For example,
dy
1)  x 2  cos x
dx
3
d3y  d2y 
2)  3   dy  e x  x sin x
dx3  dx2  dx
 
32
d2y    dy  
2

     

 dx 2    dx  
3) k 1
   

 2u  2u
4) k
x 2 t 2
 2u 2u  2u
5)  2  2 0
x 2
y z
These all are the differential equations.
Depending on the type of derivatives involved differential equations are mainly classified as
1. Ordinary Differential Equations and
2. Partial Differential Equations

Ordinary Differential Equation:
The differential equation involving derivatives or differential coefficients of dependent
variable with respect to single independent variable is called as ordinary differential equation
(ODE).
Ex. Equations 1, 2 & 3 of above are ordinary differential equations.




Dept of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur

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Partial Differential Equation:
The differential equation consisting derivatives or differential coefficients of dependent
variable with respect to two or more independent variables is called as partial differential equation
(PDE).
Ex. Equations 4 & 5 of above are partial differential equations.

Order and Degree of Differential Equation:
Order of the highest order derivative involved in the differential equation is called as order of
differential equation.
The power of the highest order derivative involved in the differential equation when it is free
from fractions and radicals as far the derivatives are concerned is called as the degree of differential
equation.
Example:
Diff. Eqn Order Degree
dy
1)  x 2  cos x 1 1
dx
3
d3y  d2y 
2)  3   dy  e x  x sin x 3 1
dx3  dx2  dx
 
 2u  2u
3) k 2 2 1
x 2
t
 u  u  2u
2 2
4)  2  2 0 2 1
x 2
y z
dy k
5) y x  1 2 (after simplifying)
dx  dy 
 
 dx 
32
d2y    dy  
2

   1   
 dx 2  
6) k 2 2 (after simplifying)
    dx  

Solution of Differential Equation:
The function or the relation between dependent and independent variables which will satisfy
the given differential equation is called as the solution of the differential equation.
For example,
c dy
1. y  (c is an arbitrary constant) is a solution of the differential equation x   y.
x dx
2
d y
2. y  e 2 x is a solution of the differential equation  4 y  0.
dx 2
Types of Solution:

General Solution:
The solution of the differential equation consisting number of arbitrary constants equal to
order of differential equation is called as its general solution.
For example,
Dept of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur

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1. y  x 2  c (c is an arbitrary constant) is a general solution of the differential equation
dy
 2x.
dx
2. y  c1 cos 2 x  c2 sin 2 x ( c1 & c2 are arbitrary constants) is general solution of the
d2y
differential equation  4 y  0.
dx 2

Particular Solution:
The solution obtained by assigning particular values to arbitrary constants appearing in
general solution of differential equation is known as its particular solution.
For example,
dy
1. y  x 2  5 is a particular solution of the differential equation  2x.
dx
d2y
2. y  2 cos 2 x  sin 2 x is the particular solution of the differential equation  4 y  0.
dx 2

1.2 Ordinary Differential Equations (ODE) of First-Order and First Degree

General Form: The general form of ordinary differential equations (ODEs) of first order and first
degree is
dy
 F ( x, y ) (1)
dx
or M ( x, y)dx  N ( x, y)dy  0 (2)

METHODS OF FINDING GENERAL SOLUTION

I. Method of Separating Variables: (Revision)
Sometime equation (1) can be written as
f ( x)dx  g ( y)dy (3)
which is said to be in variables separable form, because in (3) the variables are now separated: x
appears only on the left and y only on the right.
General Solution of equation (3) is given by
 f ( x)dx   g ( y)dy  c

EXAMPLES

Q. Solve the following differential equations.
dy 4y
1) 
dx x( y  3)
Solution: Rewriting the given equation as,
4 y 3 4  3
dx  dy or dx  1  dy …(variables separable form)
x y x  y
Integrating both sides, the required general solution is
4 log x  y  3 log y  c

Dept of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur

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dy
2)  e x  2 y  x 2e  2 y
dx
Solution: The given equation is
dy
 e x e  2 y  x 2e  2 y
dx
Rewriting it as,
e 2 y dy  (e x  x 2 )dx …(variables separable form)
Integrating both sides, the required general solution is
e2 y x x
3
e  c or 3e 2 y  6e x  2 x3  c
2 3
II. Linear Differential Equations:
A first-order ODE is said to be linear if it is of the form
dy
 P ( x) y  Q( x) (4)
dx
where P(x) and Q(x) are functions of x only or constants.
To solve linear equation first find the integrating factor by using
I .F .  e  Pdx
Then General Solution of equation (3) is given by
y( I .F .)   ( I .F .)Q( x)dx  c
Note: 1. Sometime the linear equation may be of the form
dx
 P( y ) x  Q( y )
dy
where P(y) and Q(y) are functions of y only or constants.
In this case,
I .F .  e 
Pdy
The General Solution is
x( I .F .)   ( I .F .)Q( y)dy  c
2. Variables may be different. Accordingly, use formula for I. F. and general solution.

EXAMPLES
Q. Solve the following differential equations.
dy 3
1)  y 1
dx x
dy
Solution : The given equation is linear. So, comparing with  py  Q;
dx
3
P & Q 1
x
3
 Pdx  x dy 3
I .F .  e e  e3 log x  elog x  x3
The general solution is
y( I .F .)   ( I .F .)Qdx  c

Dept of Mathematics, KIT’s College of Engineering (Autonomous), Kolhapur