Page 135 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes
Example: Solve x (x – y) dy +y dx = 0
2
Solution:

dy y 2
Here 
dx x  y x 

dy dy
v
Putting y = vx, so that   x
dx dx
dx v 2
 v  x 
dx v  1

dy v 2 v
v
 x   
dx v  1 v 1
v  1 dx
or dv 
v x

 1  1
or  1   dv  dx
 v  x
Integrating, we get
(v – log v) = log x + log c
v
or log e – log v = log cx
 e v 
or log    logcx
 v 
v
 e = vcx
y
c
or e y /x  . .x
x
1 y /x
or y  e
c
1
or y  c e y /x ; where c 1 
1 c
which is the required general solution.

2
Example: Solve (x – y ) dx = 2xy dy
2
Solution:
dy x 2  y 2
Here  ……(1)
dx 2xy

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