Page 129 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 1 1
y
x
or, x 2  xy   y 2   . C
2 2
or, x 2  y 2  2xy  2  2  C . 
y
x
Example: Solve (ey + 1)cosx dx + ey sinx dy = 0.
Solution:
The given equation is (ey + 1) cos x dx + ey sin x dy = 0 ...(1)
This is of the form Mdx + Ndy = 0
where M = (ey + 1) cos x, N = ey sin x

 M y  N y
x
  e cos ,  e cosx
y x
 M  N
since  , the equation (1) is exact.
y x
General solution is given by
 Mdx   (terms of N independent of )dy  C
x
y constant

y
 (e  1) cosxdx  0dy  C
i.e.,
y constant
or, (ey + 1) sin x = C is the general solution.

/ 
Example: Solve  1 e / x y  dx e x y  1   x  dy  0

  y
Solution:

/ 
/ x y x y  x
The given equation is  1 e  dx e  1   dy  0 ...(1)

  y
This is of the form Mdx + Ndy = 0, where
~
M = 1 + ex/y, N = ex/y (1x/y)
 M / x y  x   N / x y  x 
  e   ,   e   
y   y 2   x   y 2  

 M  N
since  , the given equation is exact.
y x
The general solution is given by

x
 Mdx   (terms of N independent of )dy  C
y constant
/ x y / x y
  (1+ e )dx   . o dy  C   y e  C
x
y constant
This is the required general solution.

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