General And Particular Solution Of Differential Equation :
Differential equation has many real-life applications such as in mechanics, Thermodynamics, wave optics, motion of object etc. Differential equation contains dependent variable independent variable and derivative of dependent variable with respect to independent variable. it is derived from cartesian equation which contains dependent variable, independent variable and constants. differential of this equation with respect to independent variable leads to differential equation.
Finding general and particular solution is a reverse process to reach the cartesian form. Now if the cartesian equation contains any arbitrary constant than this solution is known as general solution and if equation does not have any arbitrary constant than this is known as particular solution such as y =
mx
+
c
represents general cartesian equation of lines with arbitrary constants as
m
,
c
, while
y
=
x
+ 5 shows particular solution where arbitrary constants are replaced with 1, 5 respectively.
Arbitrary Constant
Arbitrary Constant.
Any cartesian equation contains variables, constants, operators and arbitrary constant. arbitrary constants contain translation, rotation, expansion, contraction of the curve such as in
y
=
mx
+
c
, arbitrary constant
m
handles rotation of the line and
c
take care of the translation part. While in equation of circle as
arbitrary constants
handles the translation part and r covers the expansion or contraction of the curve.
Differential equation of
y
=
mx
+
c
would be represented as
count of arbitrary constant shows order of differential equation, let’s explore general and particular solution of differential equation in next section with examples and illustrations.
Differential Equation Introduction
As discussed, general solution of any differential equation is defined on the basis of arbitrary constants while for any specific value of arbitrary constant particular solution is defined, let’s solve some examples
Solution Of Differential Equation Based Example
Example 1:
Verify that
is a general solution of
?
Sol.
, Now first derivative of the equation would be as
and
Since
satisfies the given differential equation hence it is the general solution of it.
Example 2:
Verify that
is a particular solution of
?
Sol.
doesn’t have any arbitrary constant hence it must be the particular solution of the given differential equation if satisfies.
first derivative of the equation would be as
and
hence verified.
Example 3:
Find general solution of the given differential equation shown as
?
Sol.
General solution of any differential equation is free from differential, hence to obtain general solution we must integrate the given differential equation.
{by parts method of integration and following ILATE Rule}
Above equation has one arbitrary constant as c.
Example 4:
Find particular solution of the given differential equation shown as
when
a
= 3 and integral constant as 0?
Sol.
Particular solution of any differential equation is free from differential and arbitrary constants hence to obtain particular solution we must integrate the given differential equation and substitute the arbitrary constant value.
{
c
= 0,
a
= 2}
Solution Of Differential Equation Based Rapid Question
1.
Find particular solution of the given differential equation shown as
when
a
= 5 and integral constant as 0?
2.
Verify that
is a general solution of
?
Illustrations based on general and particular solution Differential Equation
Q.1.
Find general solution of the given differential equation
?
Sol.
contains variable of both differentials and can be solved by separating variables.
2.
A circle having arbitrary centre as (
h
,
k
) with radius 4, find its general solution and define the differential equation?
Sol.
General equation of circle
having centre (
h
,
k
) and radius 4 would be as
… (1)
Now differential equation could be defined by differentiating it two times as it has two arbitrary constants.
From (2)
By putting both values in equation 1 differential equation would be as
Solution Of Differential Equation Based Rapid Question
1.
A parabola having arbitrary vertex as (
h
,
k
) with radius latus rectum, find its general solution and define the differential equation?
2.
Find general solution of the given differential equation
?
