The differential equation M ( x, y )dx  N ( x, y )dy  0 is called an EXACT EQUATION if there exist a continuos function F ( x, y )  c (c is a constant) such that F F dF  dx  dy x y
Excellent does not an accident, but it comes through a hard work!!

Condition for exact equation: M ( x, y )dx  N ( x, y )dy  0 is called an EXACT EQUATION if and only if M N  y x

Excellent does not an accident, but it comes through a hard work!!

GOAL
By comparing M ( x, y )dx  N ( x, y )dy  0 with F F dx  dy  dF x y

F ( x, y)  c
F M x

F N y

Test for exactness

 F M N  F    yx y x xy
2 2
Excellent does not an accident, but it comes through a hard work!!

1) Write

F F  M    (1) and  N    (2) x y

4) Compare eqn(4) with eqn(2) to obtain  ( y)

2) Integrate eqn(1) w.r.t x, so F  x dx   M dx   ( y) F ( x, y )   M dx   ( y )    (3) where  ( y ) is a function of y alone.

5) Integrate  ( y) w.r.t y to get the unknown  ( y )
6) Substitute  ( y) into eqn(3) in order to complete the required solution.

3) Differentiate eqn(3) w.r.t y F    Mdx   ( y )    (4) y y

Let’s Try

Excellent does not an accident, but it comes through a hard work!!

Solve this differential equation: (e  ye ) dx  (e  xe )dy  0; y x x y

y (1)  0

Excellent does not an accident, but it comes through a hard work!!