First Order LDE Method of Integrating factor

First Order LDE

Method of Integrating factor

Linear Differential Equations:

type-I

The standard form of a linear differential equation of first order and first degree is

where P and Q are the functions of x, or constants^.

dy

dx +P(x)y =Q(x)

Examples:

( )

1 ^dy_dx ^{+2y =6ex ;}

( )

2 ^dy_dx +ytanx =cosx ;

( )

3 ^dy_dx ^+y_x ^{=13 etc.}

The Method of Integrating Factors for Solving y'

+

p

(

x

)

y

=

q

(

x

)

.

1. Put the equation in standard form:

y' + p(x)y = q(x).

2. Calculate the integrating factor μ(x) and write it in the form

[μ(x)y]' = μ(x)q(x).

4. Integrate this equation to obtain μ(x)y = ∫μ(x)q(x) dx + c.

5. Solve for y.

Linear Differential Equations type-1

where P and Q are the functions of x, or constants.

Rule for solving dy +Py = Q dx

Integrating factor (I..F.) = ePdx

( )  

The solution is y I.F. =



Q×(IF) dx + C

Example – 1

dy x

Solve the differential equation +2y = 6e . dx

Example -2

Solve the following differential equation:

(1+x ² ) dy

dx +y =e ^tan

^-1

^x

Example –3

( )

Solve the differential equation dy+ secx y = tanx.

dx

Class/Home work

Solve the initial value problem

ty' + 2y = 4t², y(1) = 2.