First Order Differential Equations

1. Page 1

2. Page 2
Outline
 Motivation
 Basic Concepts
 Solving of First-Order Differential Equations
 Direct Integration
 Separable Differential Equations
 substitution Methods
 Linear Differential Equations

3. Page 3
Motivation

4. Basic Concepts
Page 4

5. Basic Concepts
Page 5

6. Basic Concepts
Page 6

7. First-Order Differential Equations
Page 7

8. Solving of First-Order Differential Equations
 Solving a differential equation means finding an
equation with no derivatives that satisfies the given
differential equation.
 Solving a differential equation always involves one
or more integration steps.
 It is important to be able to identify the type of
differential equation we are dealing with before we
attempt to solve it.
Page 8

9. Direct Integration
Page 9

10. Separable Differential Equations
Page 10

11. Separable Differential Equations
Page 11

12. Page 12
Exercises
Verify the solution
x
2
y
2y
xy'



13. Page 13
Explicit & Implicit Solution
 Explicit Solution
 Implicit Solution
)
(x
h
y 
0
)
,
( 
y
x
H

14. Page 14
General solution & Particular solution
,....
2
,
3
'






c
c
sinx
y
cosx
y

15. Page 15
General solution & Particular solution
kt
ce
t
y 
)
(
kt
e
t
y 2
)
( 
ky
y 


16. Page 16
Exercises
Example:
Sol:
0
4
9 

 x
y
y

17. Page 17
Exercises
Example:
Sol:
2
1 y
y 



18. Page 18
Exercises
Example:
Sol:
ky
y 


19. Page 19
Exercises
Example:
Sol:
1
)
0
(
,
2 


 y
xy
y

20. Page 20
Exercises
Substitution Method:
A differential equation of the form
can be transformed into a separable differential
equation
)
(
x
y
g
y 


21. Page 21
Substitution Method
Substitution Method:
ux
y  u
x
u
y 



x
dx
u
u
g
du
u
u
g
x
u
u
g
u
x
u










)
(
)
(
)
(

22. Page 22
Substitution Method
Example:
Sol:
2
2
2 x
y
y
xy 


cx
y
x
x
c
x
y
x
c
u
c
x
c
x
u
x
dx
u
udu
u
u
u
x
u
y
x
x
y
xy
x
xy
y
y
x
y
y
xy








































2
2
2
2
1
1
2
2
2
2
2
2
1
1
1
ln
ln
)
1
ln(
1
2
)
1
(
2
1
)
(
2
1
2
2
2

23. Page 23
Substitution Method
Exercise 1
2
01
.
0
1 y
y 


2
/
xy
y 

y
y
y
x 

 2
2
)
2
(
,
0
' 


 y
y
xy

24. Page 24
Linear Differential Equations
 Def: A first-order differential equation is said to be
linear if it can be written
 If r(x) = 0, this equation is said to be homogeneous
)
(
)
( x
r
y
x
p
y 



25. Page 25
Linear Differential Equations
 How to solve first-order linear homogeneous ODE ?
Sol: 0
)
( 

 y
x
p
y




 















dx
x
p
c
dx
x
p
c
dx
x
p
ce
e
e
e
y
c
dx
x
p
y
dx
x
p
y
dy
y
x
p
dx
dy
)
(
)
(
)
(
1
1
1
)
(
ln
)
(
0
)
(

26. Page 26
Linear Differential Equations
Example:
Sol:
0


 y
y
x
c
x
c
x
dx
dx
x
p
e
c
e
ce
ce
ce
ce
x
y
2
)
1
(
)
(
1
1
)
(












27. Page 27
Linear Differential Equations
 How to solve first-order linear nonhomogeneous ODE ?
Sol: )
(
)
( x
r
y
x
p
y 


)
(
))
(
)
(
(
)
(
1
1
0
))
(
)
(
(
)
(
)
(
x
p
x
r
y
x
p
y
Q
P
Q
dx
dF
F
dy
dx
x
r
y
x
p
x
r
y
x
p
dx
dy
x
y 














28. Page 28
Linear Differential Equations
Sol:


dx
x
p
e
x
F
)
(
)
(





 




















c
dx
r
e
e
x
y
c
dx
r
e
y
e
r
e
y
e
py
y
e
dx
x
p
dx
x
p
dx
x
p
dx
x
p
dx
x
p
dx
x
p
dx
x
p
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(

29. Page 29
Linear Differential Equations
Example:
Sol:
x
e
y
y 2



 
 
x
x
x
x
x
x
x
x
dx
dx
dx
x
p
dx
x
p
e
ce
c
e
e
c
dx
e
e
e
c
dx
e
e
e
c
dx
r
e
e
x
y
2
2
2
)
1
(
)
1
(
)
(
)
(
)
(











 








 












30. Page 30
Linear Differential Equations
Example:
)
2
cos
2
2
sin
3
(
2 x
x
e
y
y x
'




31. Page 31
Bernoulli, Jocob
Bernoulli, Jocob
1654-1705

32. Page 32
Linear Differential Equations
 Def: Bernoulli equations
 If a = 0, Bernoulli Eq. => First Order Linear Eq.
 If a <> 0, let u = y1-a
a
y
x
g
y
x
p
y )
(
)
( 


g
a
pu
a
u )
1
(
)
1
( 





33. Page 33
Linear Differential Equations
Example:
Sol:
2
By
Ay
y 



 
A
B
ce
u
y
A
B
ce
c
dx
e
A
B
e
c
dx
Be
e
u
B
Au
u
Ay
B
Ay
By
y
y
y
u
y
y
y
u
Ax
Ax
Ax
Ax
Ax
Ax
a














































1
1
)
( 1
2
2
2
1
2
1
1

34. Page 34
Linear Differential Equations
Exercise 3
4


 y
y kx
e
ky
y 



2
2 y
y
y 


1



 xy
xy
y
)
2
(
,
sin
3 
y
x
y
y 



35. Page 35
Summary
Separable 
Substitution 
Exact 
Integrating Factor 
Linear 
Bernoulli 
dx
x
f
dy
y
g )
(
)
( 
dx
x
f
du
u
g )
(
)
( 
0
)
,
(
)
,
( 
 dy
y
x
N
dx
y
x
M
0

 FQdy
FPdx
)
(
)
( x
r
y
x
p
y 


a
y
x
g
y
x
p
y )
(
)
( 

