Mathematics: Ordinary Differential Equations: First Order Linear Differential Equations

**First Order Linear
Differential Equations**

A **first order
differential equation** of the form

where *P* and *Q* are functions of *x*
only. Here no product of *dy/dx *and
its derivative *dy/dx* occur and the dependent
variable *y* and its derivative with
respect to independent variable *x*
occurs only in the first degree.

To
integrate (1), let us consider the homogeneous equation *dy/dx* + *Py*
=
0 . ….(2)

The
equation (2) can be integrated as follows:

Integrating
both sides of (3) with respect to *x*,
we get the solution of the given differential equation as

Here *e*^{∫}* ^{Pdx}* is known as the

1. The
solution of linear differential equation is

* y *×* *(*I*.*F *)*
*=* *∫*Q*(*I *.*F
*)*dx *+* C *, where* C *is an arbitrary constant.

2. In
the integrating factor *e*^{∫}
* ^{Pdx}* ,

3. A
first order differential equation of the form *dx/dy* + *Px*
=
*Q* , where *P* and *Q* are functions of *y* only. Here no product of *x* and its derivative *dx/dy* occur and the dependent variable *x *and its derivative with respect to
independent variable *y* occurs only in
the first degree.

In this
case, the solution is given by *xe*^{∫}
* ^{Pdy}* =
∫

Tags : Mathematics , 12th Maths : UNIT 10 : Ordinary Differential Equations

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12th Maths : UNIT 10 : Ordinary Differential Equations : First Order Linear Differential Equations | Mathematics

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