Linear Differential Equation :
Differential equation shows a relation between unknown functions and there derivative.it represents a set of particular family depends on the number of arbitrary constants in its Cartesian form.
Linear differential equation is defined as an equation having multiple of
y
or its derivative with respect to any independent variable as constant or functions of independent variable only such as
is a linear differential equation because multiple of y is constant and multiple of
is a function of independent variable, In general form this can be represented as
In above equation
are the terms of independent variable and can be of any degree such as
and while
is a function of x and
y
’,
y
’’,
y
’’’ are respectively the first, second, third derivative of dependent variable with respect to independent variable
Equation of the form
will not be a linear differential equation if coefficient
are function of dependent variable such as y or represents differential of dependent variable with respect to independent variable such as
Linear differential equation can be of two forms ordinary differential equation or partial differential equation
Ordinary Differential Equation
Ordinary Differential Equation:
Ordinary differential follows the above-mentioned rule and contains derivative with respect to one independent variable only.
Partial Differential Equation
Partial Differential Equation:
Partial differential follows the above-mentioned rule and contains derivative with respect to more than one independent variable such as
.
We have to discuss ordinary differential equation only.
Linear differential equation which we have to discuss is of the form
Here
denotes functions of independent variable x, let’s explore this differential equation in next section with the help of illustration and examples.
Linear Differential Equation Introduction
As discussed, one of the specific forms of linear differential equation is
To solve such type of linear differential equation we can follow the algorithm mentioned below.
1.
Find the integrating factor as I.F.=
2.
Multiply it in whole equation as
3.
Now integrate both sides with respect to x
We can directly use the result
as above steps are recursive for every question lying on the mentioned category. Let’s solve some examples on this.
Linear Differential Equation Example
Example 1:
Solve the following differential equation
Sol.
Given differential equation can be written as
Above equation represents linear differential equation compare it with
,
Find the integrating factor as
I.F.=
Use the result
Example 2:
Solve the following differential equation
Sol.
Given differential equation can be written as
Above equation represents linear differential equation compare it with
,
Find the integrating factor as
I.F.=
Use the result
Example 3:
Solve the following differential equation
given that
when
Sol.
Given differential equation can be written as
Above equation represents linear differential equation compare it with
,
Find the integrating factor as
I.F.=
Use the result
Put
Particular solution would be
Rapid Question Based on linear differential equation
1.
Solve the following differential equation
given that
when
?
2.
Solve the following differential equation
Advance Illustrations based on linear differential equation
1.
Solve the following differential equation
?
Sol.
Above equation represents linear differential equation compare it with
,
Find the integrating factor as
I.F.=
Use the result
2.
Find the particular solution of the differential equation
, given that
?
Sol.
Compare the given differential equation with
,
Find the integrating factor as
I.F.=
Use the result
Put
Particular solution would be as
1.
Find the general solution of the differential equation
and also find the equation of curve at the point (1, 1)?
2.
Find the particular solution of the differential equation
, given that
?
