Exact Differential Equation

2.  Equations which are composed of an unknown
function and its derivatives are called differential
equations.
 Differential equations play a fundamental role in
engineering because many physical phenomena are
best formulated mathematically in terms of their rate
of change.
v- dependent variable
t- independent
variable
v
m
c
g
dt
dv


3.  When a function involves one dependent variable, the
equation is called an ordinary differential equation
(or ODE). A partial differential equation (or PDE)
involves two or more independent variables.
 Differential equations are also classified as to their
order.
 A first order equation includes a first derivative as its
highest derivative.
 A second order equation includes a second derivative.
 Higher order equations can be reduced to a system of
first order equations, by redefining a variable.

4.  Consider a first order ODE of the form
 Suppose there is a function  such that
and such that (x,y) = c defines y = (x) implicitly.
Then
and hence the original ODE becomes
 Thus (x,y) = c defines a solution implicitly.
 In this case, the ODE is said to be exact.
0),(),(  yyxNyxM
),(),(),,(),( yxNyxyxMyx yx  
 )(,),(),( xx
dx
d
dx
dy
yx
yyxNyxM 








  0)(, xx
dx
d


5.  Suppose an ODE can be written in the form
where the functions M, N, My and Nx are all continuous in
the rectangular region R: (x, y)  (,  ) x (,  ). Then
Eq. (1) is an exact differential equation iff
 That is, there exists a function  satisfying the conditions
iff M and N satisfy Equation (2).
)1(0),(),(  yyxNyxM
)2(),(),,(),( RyxyxNyxM xy 
)3(),(),(),,(),( yxNyxyxMyx yx  

6.  It is sometimes possible to convert a differential equation
that is not exact into an exact equation by multiplying the
equation by a suitable integrating factor (x,y):
 For this equation to be exact, we need
 This partial differential equation may be difficult to solve. If
 is a function of x alone, then y = 0 and hence we solve
provided right side is a function of x only. Similarly if  is a
function of y alone. See text for more details.
0),(),(),(),(
0),(),(


yyxNyxyxMyx
yyxNyxM

      0  xyxyxy NMNMNM
,

N
NM
dx
d xy 


8.  
2
2
A non-homogeneous second order differential equation is of the form
d y dy
a b cy f x
dx dx
  
We find the general solution of the homogeneous
equation as before
2
2
The general solution to the equation
0
is known as the complementary function.
d y dy
a b cy
dx dx
  
Step 1

9. Step 2
Find a particular solution to the non homogeneous equation.
This solution is called the particular integral and looks
similar to f(x)
Step 3
The general solution of the non-homogeneous differential
equation is the sum of the complementary function and
the particular integral

11.   2 1 particular integral isf x x y px q    
 
 
2 2
2
1 particular integral is
: 0 1
f x x y px qx r
note f x x x
     
  
  2 2
4 particular integral isx x
f x e y pe  
  2sin cos particular integral is sin cosf x x x y p x q x    
 
 
3sin 2 particular integral is sin 2 cos2
: 3sin 2 0cos2
f x x y p x q x
note f x x x
   
 

12. 2
2
Find the general solution of the second
order differential equation
2 4 1
d y dy
y x
dx dx
   
  
2
2
2 0
2 1 0
2 and 1
x x
k k
k k
k k
y Ae Be
  
  
  
 
Complementary Function

13. Particular Integral
2
2
and 0
y px q
dy d y
p
dx dx
 
 
     
2
2
2 4 1
0 2 4 1
2 2 4 1
2 2 4 1
d y dy
y x
dx dx
p px q x
p px q x
px p q x
   
    
    
     equate coefficients
4 2
2 4 and 2
x px
p p
 
   
 
1 2
2 2 1
2 2 1
1
2
p q
q
q
q
  
   
 


14. The general solution of the non-homogeneous differential
equation is the sum of the complementary function and
the particular integral
2 1
2
2
x x
y Ae Be x
   
1
with 2 and
2
1
2
2
y px q p q
y x
    
  