Page 149 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes
d Pdx  Pdx
i.e.,  ye   Qe
dx
Integrating, we have

ye Pdx  Qe Pdx dx  , …….(2)
c
which is the required general solution.

Notes

1. e Pdx is known as Integrating factor, in short, I.F.

2. Linear differential equation is commonly known as Leibnitz’s linear equation.

Example:
dy
y
Solve cosx  sin x  1
dx
Solution:
Given equation can be written as

dy
y
 tanx  secx ….(1)
dx
Here P = tan x, Q = sec x.

F
 I . .  e tanxdx  e logsecx  sec .
x
 Solution of (1) is
x
y . secx sec . secxdx c
c
or y secx tan x  . Ans.

Example:

2 dy 2
Solve  1 x   2xy 4x  0.
dx
Solution:
Given equation is
dy 2x 4x 2
 y  . ….(1)
dx 1 x 2 1 x 2

2x 4x 2
Here P  2 , Q  2 .
1 x 1 x

2x
 dx 2
 I . .  e Pdx  e 1 x 2  e log (1 x )  (1 x 2 ).

F
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