Summary Ordinary Differential Equations, ISBN: 9780486649405 Engineering maths

Differential
Equations
Differential Equation
An equation that involves an independent variable, dependent variable
and differential coefficients of dependent variable with respect to the
independent variable is called a differential equation.
 d 2 y  dy 
3
e.g. (i) x 2  2  + x3   = 7x 2 y 2
 dx   dx 

(ii) ( x 2 + y 2 ) dx = ( x 2 − y 2 ) dy

Order and Degree of a Differential Equation
The order of a differential equation is the order of the highest
derivative occuring in the equation. The order of a differential equation
is always a positive integer.
The degree of a differential equation is the exponent of the derivative
of the highest order in the equation, when the equation is a polynomial
in derivatives, i.e. in y ′ , y ′′ , y ′′′ etc.
e.g. The order and degree of a differential equation
2 3
 d3 y   d 2 y
 3  + 2  2  + 3 y = 0 are 3 and 2 respectively.
 dx   dx 
Note If the differential equation is not a polynomial equation in derivatives,
then its degree is not defined.
dy  dy 
e.g. Degree of + cos   = 0 is not defined,
dx  dx 
dy  dy 
as + cos   = 0 is not a polynomial in derivatives.
dx  dx 

, Linear and Non-Linear Differential Equations
A differential equation is said to be linear, if the dependent variable
and all of its derivatives occuring in the first power and there are no
product of these.
A linear equation of nth order can be written in the form
dn y dn − 1y dn − 2y dy
P0 + P1 n −1
+ P2 + K + Pn − 1 + Pn y = Q
dx n
dx dx n − 2 dx
where, P0 , P1 , P2 , K , Pn − 1, Pn and Q must be either constants or
functions of x only.
A linear differential equation is always of the first degree but every
differential equation of the first degree need not be linear.
2
d2y  dy  d2y dy
e.g. The equations +   + xy = 0, x +y + y = x3
dx 
2
dx  dx 2
dx
2
 dy  d y
and   + y = 0 are not linear.
 dx  dx 2


Solution of Differential Equations
A solution of a differential equation is a relation between the variables,
of the equation not involving the differential coefficients, such that it
satisfy the given differential equation (i.e., from which the given
differential equation can be derived).
d2y
e.g. y = A cos x + B sin x is a solution of + y = 0, because it satisfy
dx 2
this equation.

1. General Solution
If the solution of the differential equation contains as many
independent arbitrary constants as the order of the differential
equation, then it is called the general solution or the complete integral
of the differential equation.
d2y
e.g. The general solution of + y = 0 is y = A cos x + B sin x because
dx 2
it contains two arbitrary constants A and B, which is equal to the order
of the equation.