The document provides an overview of differential equations, defining key concepts such as ordinary and partial differential equations, their orders and degrees, as well as methods for solving first-order separable, homogeneous, and exact equations. It explains the forms these equations take and outlines various solution methods, including the use of integrating factors. Examples are given for clarification, emphasizing the importance of different techniques in solving first-order differential equations.

1. PRESENTED BY
ITISHREE DASH
ASSISTANT PROFESSOR
DEPARTMENT OF BSH

2. Differential Equations
DIFFERENTIAL EQUATION
An equation which involves derivatives of one or more dependent
variables w.r.t one or more independent variable
 ordinary differential equation (ode) : D.E in which the
dependent variable y depends only on one independent
variable say x.
 partial differential equation (pde) : D.E in which
dependent variable y depends on two or more
independent variables say x , t etc.

3. ORDER OF D.E : Order of differential equation is the order of the
highest order derivative appearing in the equation.
DEGREE OF D.E : Degree of differential equation is the degree of the
highest ordered derivatives.
LINEAR D.E : An D.E the is said to be linear in if
(i) Every dependent variable and derivatives involved occurs to the first
degree only and
(ii) Products of derivatives and/or dependent variables do not occur.
NOTE : D.E which is not linear is called non-linear differential equation.
Differential Equations

4. Differential Equations

5. 
Differential Equations

6. Differential Equations
 A linear differential equation of order n is a differential equation
written in the following form:
       

 
    
1
1 1 01
( )
n n
n nn n
d y d y dy
a x a x a x a x y f x
dx dx dx
where an(x) is not the zero function
 General solution : looking for the unknown function of a
differential equation
 Particular solution (Initial Value Problem) : looking for
the unknown function of a differential equation where the
values of the unknown function and its derivatives at some
point are known

7. 1st Order DE - Separable Equations
The differential equation M(x,y)dx + N(x,y)dy = 0 is separable if
the equation can be written in the form:
        02211  dyygxfdxygxf
Solution :
1. Multiply the equation by integrating factor:
   ygxf 12
1
2. The variable are separated :
 
 
 
 
0
1
2
2
1
 dy
yg
yg
dx
xf
xf
3. Integrating to find the solution:
 
 
 
 
Cdy
yg
yg
dx
xf
xf
  1
2
2
1

8. 1st Order DE - Separable Equations
Examples:
1. Solve : dyxdxydyx 2
4 
Answer:

9. 1st Order DE - Separable Equations
Examples:
1. Solve :
y
x
dx
dy 22


Answer:

10. 1st Order DE - Separable Equations
Examples:
2. Find the particular solution of :   21;
12


 y
x
y
dx
dy
Answer:

11. 1st Order DE - Homogeneous Equations
Homogeneous Function
f (x,y) is called homogenous of degree n if :    y,xfy,xf n
 
Examples:
  yxxy,xf 34
  homogeneous of degree 4
       
   yxfyxx
yxxyxf
,
,
4344
34




  yxxyxf cossin, 2
  non-homogeneous
       
   
 yxf
yxx
yxxyxf
n
,
cossin
cossin,
22
2







12. 1st Order DE - Homogeneous Equations
The differential equation M(x,y)dx + N(x,y)dy = 0 is homogeneous
if M(x,y) and N(x,y) are homogeneous and of the same degree
Solution :
1. Use the transformation to : dvxdxvdyvxy 
2. The equation become separable equation:
    0,,  dvvxQdxvxP
3. Use solution method for separable equation
 
 
 
 
Cdv
vg
vg
dx
xf
xf
  1
2
2
1
4. After integrating, v is replaced by y/x

13. 1st Order DE – Homogeneous Equations
Examples:
1. Solve :   03 233
 dyxydxyx
Answer:

14. 1st Order DE - Homogeneous Equations
Examples:
2. Solve :
2
52
yx
yx
dx
dy



Answer:

15. 1st Order DE – Exact Equation
The differential equation M(x,y)dx + N(x,y)dy = 0 is an exact
equation if :
Solution :
The solutions is given by the equation
x
N
y
M





  CyxF ,
1. Integrate either M(x,y) with respect to x or N(x,y) to y.
Assume integrating M(x,y), then :
where : F/ x = M(x,y) and F/ y = N(x,y)
     ydxyxMyxF   ,,
2. Now :       yxNydxyxM
yy
F
,', 





 
or :       

 dxyxM
y
yxNy ,,'

16. 1st Order DE – Exact Equation
3. Integrate ’(y) to get (y) and write down the result F(x,y) = C
Examples:
1. Solve :     01332 3
 dyyxdxyx
Answer:

17. 1st Order DE – Exact Equation
Examples:
2. Solve :   0cos214 2

dx
dy
yxxy
Answer:

18. 1st Order DE – Non Exact Equation
The differential equation M(x,y)dx + N(x,y)dy = 0 is a non exact
equation if :
Solution :
The solutions are given by using integrating factor to change the
equation into exact equation
x
N
y
M





1. Check if :   onlyxoffunctionxf
N
x
N
y
M

 



then integrating factor is
  dxxf
e
  onlyyoffunctionyg
M
y
M
x
N

 



or if :
then integrating factor is
  dyyg
e

19. 1st Order DE – Non Exact Equation
2. Multiply the differential equation with integrating factor which
result an exact differential equation
3. Solve the equation using procedure for an exact equation

20. 1st Order DE – Non Exact Equation
Examples:
1. Solve :   022
 dyxydxxyx
Answer:

21. 1st Order DE – Non Exact Equation
Examples:
2. Solve :
Answer:

22. I. 1st order differential equation can be solved using
variable separation method.
II. By checking whether the equation is homogeneous
or exact we can solve according to the procedure.
Differential Equations

23. THANK YOU