Chapter: 1
Differential Equation Basics:
Definition: An equation involving independent variable, dependent variable
and differential coefficient of dependent variable with respect to independent
variable, is called a differential equation. An equation involving one or more
differentials or differentials coefficients is called differential equation. That is,
the variables may or may not lie in differential equation but the differential
coefficient must lie in differential equation.
Let x is called independent variable and y is called dependent variable. A
derivative of y with respect to x is denoted by
𝑑𝑦
𝑑𝑥
. If Y(x) is function is a
function of x then its derivative is also denoted by Y ′.This is called first
derivative of Y with respect to x. If it is further differentiated with respect to x
then is denoted by Y″ and it is called 2nd derivative of Y with respect to x.
Similarly Y‴, Y‴′, Y‴″ etc. The 2nd, 3rd, etc. derivatives are generally
𝑑2𝑦 𝑑3y
represented in mathematics by
,
d𝑥 2 d𝑥 3
respectively.
Here, dy means an infinitesimal change in “y”
dx means an infinitesimal change in “x”
Various types of differential equation
 Mainly there are 2 types of differential equation.
1) Ordinary differential equation
2) Partial differential equation.
Ordinary differential equation:
An equation involving one independent variable, one dependent variable and
derivative of dependent variable with respect to a single independent variable
is called an ordinary differential equation.
𝑑2𝑦
Example: 1.
d𝑥 2
𝑑𝑦
-5
𝑑𝑥
𝑑2𝑦
2. 𝑥 2
d𝑥 2
+6y=0
+ 2x
𝑑𝑦
𝑑𝑥
+y=0
Partial differentiation equation:
An equation involving one dependent variable, two or more independent
variables and partial differential coefficient of dependent variable with
respect to more than one independent variable is called partial differentiation
equation.
Example: 1. x
𝜕𝑧
𝜕𝑥
2.
𝜕2 z
+y
+
𝜕𝑥 2
𝜕𝑧
𝜕𝑦
𝜕2 z
𝜕𝑦 2
=3z
=0
Order of a differential equation:
The order of the highest order differential coefficient involved in a differential
equation is called the order of the differential equation.
Example: 1.
2.
𝑑2𝑦
d𝑥 2
𝑑𝑦
+3
𝑑𝑥
𝑑4x
𝑑2𝑦
d𝑦
d𝑥 2
+7
4
+2y=0
+5y=𝑐𝑜𝑠 2 𝑥
In the above, the equation (1) is of the second order and the equation (2) is of
the fourth order.
Degree of a differential equation:
The degree or power of the highest order derivative of a differential equation
is called the degree of a differential equation, when the equation has been
made free from fraction.
Example: 1.
2.
𝑑3y
d𝑥 3
𝑑2𝑦
d𝑥 2
𝑑𝑦
+3( ) 4 +y=0
𝑑𝑥
𝑑𝑦
=k[1 + ( ) 2 ]5/3
𝑑𝑥
In the equation (1), the power of the highest third order derivative is 1. So the
degree of equation is one. If so the power of
𝑑𝑦
𝑑𝑥
is 4 but
𝑑𝑦
𝑑𝑥
is not the highest
order differential coefficient. So the degree of the equation is not 4. Hence the
degree of equation is 1.
If eliminating the fractional power of equation (2) then the equation is of the
form
(
𝑑2𝑦 3
) =
d𝑥 2
𝑑𝑦
𝑘 3 [1 + ( ) 2 ]5
𝑑𝑥
In this case, the power of the highest second order derivative is 3. Hence the
degree of equation is 3.
Solution of differential equation:
If any relation between dependent and independent variables is determined
from the given differential equation which satisfies the given differential
equation then this differential equation is called its solution.
Example: 1.
𝑑𝑦
𝑑𝑥
=2x
Integrating (1) with respect to x we get,
y=𝑥 2 +c; when c is an arbitrary constant.
Hence the solution of (1) is y=𝑥 2 +c;
Solutions of differential equations are generally categorized by two types.
These are:
 General or Complete solution.
 Particular solution.
General or Complete solution:
The solution of a differential equation of n-th order contains n arbitrary
constant which satisfies the given differential equation is called its general or
complete solution.
Example: 1. (
𝑑2𝑦
d𝑥 2
𝑑𝑦
)- 5 +6y=0
𝑑𝑥
Solution of (1) is y=𝑐1 𝑒 2𝑥 +𝑐2 𝑒 3𝑥 …………………(2)
Where 𝑐1 and 𝑐2 are arbitrary constants. Since equation (1) is satisfied by (2)
and the solution (2) have two arbitrary constants. So equation (2) is called the
general or complete solution.
Particular solution:
In a particular case or condition a solution is obtained for the definite values
to the arbitrary constants of general solution is called a particular solution.
Example: (
𝑑2𝑦
d𝑥 2
𝑑𝑦
)- 5 +6y=0……………………………(1)
𝑑𝑥
General solution of (1) is y=𝑐1 𝑒 2𝑥 +𝑐2 𝑒 3𝑥 …………………(2)
If in any particular case or condition 𝑐1 =2 and 𝑐2 = 1; then the particular
solution of (1) is y=2𝑒 2𝑥 +𝑒 3𝑥