Transcript differential equation

First-Order Differential Equations
S.-Y. Leu
Sept. 28, 2005
1
CHAPTER 2
First-Order Differential Equations
2.1 Solution Curves Without the Solution
2.2 Separable Variables
2.3 Linear Equations
2.4 Exact Equations
2.5 Solutions by Substitutions
2.6 A Numerical Solution
2.7 Linear Models
2.8 Nonlinear Models
2.9 Systems: Linear and Nonlinear Models
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2.1 Solution Curves Without the Solution
y
Slope=
dy
 y'
dx
x
Short tangent segments suggest the
shape of the curve
輪廓
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2.1 Solution Curves Without the Solution

The general first-order differential equation has
the form
F(x, y, y’)=0
or in the explicit form
y’=f(x,y)


Note that, a graph of a solution of a first-order
differential equation is called a solution curve
or an integral curve of the equation.
On the other hand, the slope of the integral
curve through a given point (x0,y0) is y’(x0).
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2.1 Solution Curves Without the Solution


A drawing of the plane, with short line
segments of slope drawn at selected points ,
is called a direction field or a slope field of
the differential equation .
The name derives from the fact that at each
point the line segment gives the direction of
the integral curve through that point. The
line segments are called lineal elements.
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2.1 Solution Curves Without the Solution
Plotting Direction Fields

1st Step
y’=f(x,y)=C=constant curves of equal inclination

2nd Step
Along each curve f(x,y)=C, draw lineal elements
 direction field

3rd Step
Sketch approximate solution curves having the
directions of the lineal elements as their tangent
directions.
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2.1 Solution Curves Without the Solution

If the derivative dy/dx is positive
(negative) for all x in an interval I,
then the differentiable function y(x)
is increasing (decreasing) on I.
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2.1 Solution Curves Without the Solution
DEFINITION: autonomous DE
A differential equation in which the independent
variable does not explicitly appear is known
as an autonomous differential equation.
For example, a first order autonomous DE has
the form y '  f ( y)
DEFINITION: critical point
'
A critical point of an autonomous DE y  f ( y) is a real
number c such that f(c) = 0.
Another name for critical point is stationary point or
equilibrium point.
If c is a critical point of an autonomous DE, then y(x) = c is a
constant solution of the DE.
2.1 Solution Curves Without the Solution
DEFINITION: phase portrait
A one dimensional phase portrait of an
'
y
 f ( y) is a diagram which indicates
autonomous DE
the values of the dependent variable for which y is
increasing, decreasing or constant.
Sometimes the vertical line of the phase portrait is
called a phase line.
2.2 Separable
Variables
DEFINITION: Separable DE
A first-order differential equation of
the form
dy
 g ( x ) h( y )
dx
is said to be separable or to
have separable variables.
(Zill, Definition 2.1, page 44).
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2.2 Separable Variables
Method of Solution
dy
 g ( x ) h( y )
dx

1 dy
dy
 p( y )
 g ( x)
h( y ) dx
dx
If y   (x) represents a solution
'
p
(

(
x
))

( x)  g ( x)

'

p( ( x)) ( x) dx  g ( x) dx




p ( y ) dy   g ( x) dx
H ( y)  c1  G( x)  c2
dy   ' ( x) dx
H ( x)  G ( x)  c
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2.2 Separable Variables
The Natural Logarithm
x
1
domain
,
ln x 
dt
1
t

,
ln x  log e x
1
Dx ln x 
x
ln( ab)  ln a  ln b
ln  (0, )
e  2.71828
1
D ln x   2
x
2
x
ln
a
 ln a  ln b
b
ln a r  r ln a
The natural exponential function
domain exp  (, )
exp x  e x
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2.3 Linear
Equations
DEFINITION: Linear Equation
A first-order differential equation of
the form
dy
a1 ( x)
 a0 ( x ) y  g ( x)
dx
is said to be a linear equation.
(Zill, Definition 2.2, page 51).
When g ( x)  0 homogeneous
Otherwise it is non-homogeneous.
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2.3 Linear
Equations
Standard Form
a1 ( x)

dy
 a0 ( x ) y  g ( x)
dx
dy
 p( x) y  f ( x)
dx
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2.2 Separable Equations


A differential equation is called separable if it
can be written as y '  A( x) B( y )
Such that we can separate the variables and
write
1
B( y )

dy  A( x)dx
B( y )  0
We attempt to integrate this equation

1
dy 
B( y )

A( x)dx
2.2 Separable Equations

Example 1.
y ye
'
as
2
x
dy
x

e
dx
2
y
is separable. Write
dy
 y 2ex
dx
y0
Integrate this equation to obtain
or in the explicit form y  1
1
  e  x  k
y
ex  k
What about y=0 ? Singular solution !
2.2 Separable Equations

Example 2.
x 2 y '  1  y is separable, too. We write
dy
dx
 2
1 y x
x0
y  1
Integrate the separated equation to obtain
1
ln 1  y    k
1  y  e k e 1 / x  Ae 1 / x
x
 1  y   Ae 1 / x  Be 1 / x
y  1  Be 1 / x
The general solution is
Again, check if y=-1 is a solution or not ?
it is a solution, but not a singular one, since it is a special case of the
general solution
2.3 Linear Differential Equations


A first-order differential equation is linear if it has the
form y ' ( x)  p ( x) y  q ( x)
 p ( x ) dx
Multiply the differential equation by e
to get
e
p ( x ) dx
y ' ( x)  p( x)e 
d
 y ( x )e 
dx

  q ( x )e
y  q( x)e 
y ( x)  e  
p ( x ) dx
p ( x ) dx
 p ( x ) dx
Now integrate to obtain y( x)e


p ( x ) dx
p ( x ) dx
 q( x)e
p ( x ) dx
 p ( x ) dx

 q( x)e
dx  C e
 p ( x ) dx
dx  C
  p ( x ) dx
The function e  p ( x ) dx is called an integrating factor for
the differential equation.
2.3 Linear Differential Equations



Linear: A differential equation is called linear if
there are no multiplications among
dependent variables and their derivatives.
In other words, all coefficients are functions of
'
( x)  p ( x) y  q( x)
independentyvariables.
Non-linear: Differential equations that do not
satisfy the definition of linear are non-linear.
Quasi-linear: For a non-linear differential
equation, if there are no multiplications among
all dependent variables and their derivatives in
the highest derivative term, the differential
equation is considered to be quasi-linear.
2.3 Linear Differential Equations

Example
y '  y  sin( x)
is a linear DE. P(x)=1 and
q(x)=sin(x), both continuous for all x.
An integrating factor is e p ( x ) dx  e dx  e x
Multiply the DE by e
Or  ye x '  e x sin( x)
Integrate to getye 
x

to get y e
x
'
x
 ye  e sin( x)
x
x
1
e sin( x) dx  e sin( x)  cos(x)  C
2
The general solution is
x
x
y
1
sin( x)  cos(x)  Cex
2
2.3 Linear Differential Equations
Example
y
'
2
Solve the initial value problem y  3x  x ; y (1)  5
It can be written in linear form ' 1
2

y 
x
y  3x
x0
An integrating factor is e  (1/ x ) dx  eln(x )  x
for
Multiply the DE by x
to get xy '  y  3x 3
Or xy '  3x 3
3 3 C
3 4
y x 
Integrate to get xy  x  C
,then
4
x
4
for x  0
3
For the initial condition, we need y(1)  5   C
4
C=17/4 the solution of the initial value problem
3 3 17
is
y ( x)  x 
4
4x