Page 125 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes can be reduced to a form in which the variables are separable by the substitution ax + by + c = u
dy du
so that a  b 
dx dx

dy 1  du 
or    a 
dx b  dx 
Equation (1) reduces to

1  du 
  a      
b  dx 
1  du 
or   a      
b  dx 
du
or  dx .
b
a     u
After integrating both sides, u is replaced by its values.


x
x
2
Example: Solve 3e tan y dx + (1 + e ) sec y dy = 0, given y  when x = 0
4
Integrating, we get
3log  1 e x   log tan y   log ,c
3
c
or log   1 e x  tan y   log ,
   
x 3
or (1 +e ) tan y = c. …(1)
which is the general solution of the given equation.

Since y when x = 0, we have from (1),
4
(1 + 1)  1 = c
3
 c = 8
 The required particular solution is
(1 + e ) tan y = 8.
x 3

Example:

dy x 2logx   1
Solve 
y
dx siny  cosy
Solution:

The given equation can be re-written as
(sin y + y cos y) dy = x (2 log x + 1) dx

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