??? Methods of Solving the First Order Differential Equation

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??? Methods of Solving the First Order
Differential Equation
graphic method
direct integration
numerical method
separable variable
analytic method
method for linear equation
method for exact equation
homogeneous equation method
Bernoullis equation method
method for Ax By c
series solution
Laplace transform
matrix solution
Fourier series
Fourier cosine series
transform
Fourier sine series
Fourier transform
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Simplest method for solving the 1st order DE
Direct Integration dy(x)/dx f(x)
where
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??? Table of Integration
1/x lnx c
cos(x) sin(x) c
sin(x) cos(x) c
tan(x) lncos(x) c
cot(x) lnsin(x) c
ax ax/ln(a) c



x eax
x2 eax
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Others about Integration
(1) Integration ???
?
(2) ?? integration ??????? constant c
(3) If
then
c1 is also some constant
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2-2 Separable Variables
2-2-1 ???????
1st order DE ????? dy(x)/dx f(x, y)
Definition 2.2.1 (text page 46) If
dy(x)/dx f(x, y) and f(x, y) can be separate
as f(x, y)
g(x)h(y) i.e., dy(x)/dx
g(x)h(y) then the 1st order DE is separable (or
have separable variable).
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dy(x)/dx g(x)h(y)
??
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2-2-2 ??
If ,
then Step 1
where p(y)
1/h(y) Step 2

where
????
????
Extra Step (a) Initial conditions
(b) Check the singular solution (i.e., the
constant solution)
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Extra Step (b) Check the singular solution
(???)
Suppose that y is a constant r
solution for r
See whether the solution is a special case of the
general solution.
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2-2-3 Examples
Example 1 (text page 47) (1 x) dy y dx
0
Extra Step (b)
check the singular solution
Step 1
set y r , 0 r/(1x) r 0, y 0
Step 2
(a special case of the general solution)
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Example ????? ???????,??????? (?????????????) Exe
rcise ????? ???,???????????? ???????????,?????????
?? ????????,????????????? ?????,???
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Example 2 (with initial condition and implicit
solution, text page 48)
, y(4) 3
Extra Step (b)
check the singular solution
Step 1
Step 2
Extra Step (a)
(implicit solution)
invalid
valid
(explicit solution)
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Example 3 (with singular solution, text page 48)
Extra Step (b)
check the singular solution
Step 1
set y r , 0 r2 4 r ?2, y ?2
Step 2
or
y ?2
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Example 4 (text page 49) ?? ??????
,
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Example in the top of page 50
, y(0) 0
Extra Step (b) Check the singular solution
Step 1
Step 2
Extra Step (a)
Solution or
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????, ?????????
, y(0) 0
solutions (1) (2)
(3)
b ? 0 ? a
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2-2-4 IVP ???????
???????????(Theorem 1.2.1, text page 16)
?? f(x, y), ? x x0, y y0 ????
continuous
??????? h,?? IVP ? x0-h lt x lt x0 h ?????????
?????
J. Ratzkin, Existence and Uniqueness of Solutions
to First Order Ordinary Differential Equations,
2007.
The Existence and Uniqueness Theorem for
First-Order Differential Equations,
www.math.uiuc.edu/tyson/existence.pdf
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2-2-5 Solutions Defined by Integral
(1) (2) If dy/dx g(x) and y(x0) y0, then

?? (integral, antiderivative) ?????
function, ???? nonelementary
? ,
??,solution ?????
???
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Example 5 (text page 50)
Solution
??????? complementary error function
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? error function (useful in probability)
? complementary error function
? t ?? x ????
See text page 59 in Section 2.3
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2-2-6 ????????
(1) ?????????????? (2) ???? c ????????? c
??????? (3) ?????,????? singular solution (4)
???,??????
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??? ??????
http//integrals.wolfram.com/index.jsp
?????,??????????
??
(a) ??integrals.wolfram.com/index.jsp ????
(b) ????????????,??
???
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(c) ??? Compute Online with Mathematica
??????????
?
??
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(d) ??,???????????,??????,?????????????
??
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???????
http//mathworld.wolfram.com/
???????????????????
http//www.sosmath.com/tables/tables.html
?????? mathematical table (???????)
http//www.seminaire-sherbrooke.qc.ca/math/Pierre/
Tables.pdf
?????? mathematical table,?? convolution, Fourier
transform, Laplace transform, Z transform
????, Maple, Mathematica, Matlab ????????????
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2-3 Linear Equations
friendly form of DEs
2-3-1 ???????

• Definition 2.3.1 The first-order DE is a
  linear equation if it has the following form
• g(x) 0 homogeneous
• g(x) ? 0 nonhomogeneous

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Standard form
????????,?????? linear first order DE
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2-3-2 ?????
??? 1
??? 2
Find the general solution yc(x) (homogeneous
solution)
Find any solution yp(x) (particular solution)
Solution of the DE

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? yc yp is a solution of the linear first order
DE, since

• Any solution of the linear first order DE should
  have the form yc yp .
• The proof is as follows. If y is a solution of
  the DE, then

Thus, y - yp should be the solution of
y should have the form of y yc yp
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Solving the homogeneous solution yc(x) (????)
separable variable
Set , then
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Solving the particular solution yp(x)
(????) Set yp(x) u(x) y1(x) (?? particular
solution ? homogeneous solution ??????)
equal to zero
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solution of the linear 1st order DE
where c is any constant
integrating factor

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2-3-3 ??
(Step 1) Obtain the standard form and find P(x)
(Step 2) Calculate (Step 3a) The standard form
of the linear 1st order DE can be rewritten
as (Step 3b) Integrate both sides of the above
equation
remember it
or remember it, skip Step 3a
(Extra Step) (a) Initial value
(c) Check the Singular Point
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Singular points the locations where a1(x) 0
i.e., P(x) ? ? More
generally, even if a1(x) ? 0 but P(x) ? ? or f(x)
? ?, then the location is also treated as a
singular point. (a) Sometimes, the solution may
not be defined on the interval including the
singular points. (such as Example 4) (b)
Sometimes the solution can be defined at the
singular points, such as Example 3
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More generally, even if a1(x) ? 0 but P(x) ? ? or
f(x) ? ?, then the location is also treated as a
singular point.
Exercise 33
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2-3-4 ??
Example 2 (text page 56)
Extra Step (c)
Step 1
check the singular point
Step 2
???????? 3xc ??? 3x?
Step 3
??,?? Step 3,?????
Step 4
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Example 3 (text page 57)
Extra Step (c)
check the singular point
Step 1
x 0
Step 2
?? x lt 0 ???
???? x gt 0 ???,
Step 3
Step 4
x ??? (0, ?)
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Example 4 (text page 58)
Extra Step (c)
check the singular point
defined for x ? (?, 3), (3, 3), or (3, ?) not
includes the points of x 3, 3
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Example 6 (text, page 59)
check the singular point
0 ? x ? 1
x gt 1
?? y(x) ? x 1 ???? continuous
from initial condition
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2-3-5 ?????
(1) transient term, stable term
Example 5 (text page 58) ???
transient term ? x ?????? x ?1 stable term
y
x?1
x-axis
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(2) piecewise continuous A function g(x) is
piecewise continuous in the region of x1, x2
if g'(x) exists for any x ? x1, x2.
In Example 6, f(x) is piecewise continuous in
the region of 0, 1) or (1, ?)
(3) Integral (??) ?????? antiderivative
(4) error function
complementary error function
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(5) sine integral function
Fresnel integral function
(6)
f(x) ???? input ? driving function
Solution y(x) ???? output ? response
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2-3-6 ???
When is not easy to calculate Try
to calculate
Example
(not linear, not separable)
(linear)
(implicit solution)
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2-3-7 ????????
(1) ??? linear 1st order DE ?? standard form (2)
??? singular point
??singular point ? Section 2-2 ??? singular
solution ??
(3) ????
?
(4) ???, ????????
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???????,????
??? realize remember it ?? realize it ??
remember it ?? read it without
realization and remembrance ??? rest
z..z..z
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Chapter 3 Modeling with First-Order Differential
Equations
???

• Convert a question into a 1st order DE.
• ?????????
• (2) Many of the DEs can be solved by
• Separable variable method or
• Linear equation method
• (with integration table remembrance)

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3-1 Linear Models
Growth and Decay (Examples 13) Change the
Temperature (Example 4) Mixtures (Example
5) Series Circuit (Example 6)
??? Section 2-3 ?????
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Example 1 (an example of growth and decay,
text page 84) Initial A culture (???)
initially has P0 number of bacteria. The other
initial condition At t 1 h, the number of
bacteria is measured to be 3P0/2. ??? If the
rate of growth is proportional to the number of
bacteria A(t) presented at time t, Question
determine the time necessary for the number of
bacteria to triple
?? ? A(0) P0
?? ? A(1) 3P0/2
?? ? k is a constant
?? ? find t such that A(t) 3P0
?????? P(t) ?? A(t)
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A(0) P0, A(1) 3P0/2
??? ??????
Extra Step (b)
check singular solution
Step 1
Step 2
Extra Step (a)
(1) c P0 (2)
k ln(3/2) 0.4055
????????
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??? linear (Section 2.3) ????? Example 1
??????????? initial values ?????????
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Example 4 (an example of temperature change,
text page 86) Initial When a cake is removed
from an oven, its temperature is measured at 149?
C.
?? ? T(0) 149
The other initial condition Three minutes later
its temperature is 85 ? C.
?? ? T(3) 85
question Suppose that the room temperature is
21? C. How long will it take for the cake to
cool off to 22? C? (??????????????)
?? ? find t such that T(t) 22.
??,????,?????????????????????????,?? T(t) ???? DE
????
k is a constant
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T(0) 149
T(3) 85
??? separable variable ???? ??? linear ??????
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Example 5 (an example for mixture, text page 87)
Concentration 0.24 kg/L
1135 L (liters)
11.35 L/min
11.35 L/min
A the amount of salt in the tank
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LR series circuit
From Kirchhoffs second law
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RC series circuit
q ??
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How about an LRC series circuit?
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Example 7 (text page 89) LR series circuit ?
E(t) 12 volt, ? inductance 1/2 henry, ?
resistance 10 ohms, ? initial current 0
?? c1 ???
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Circuit problem for t is small and t
?
For the LR circuit L R
transient stable
For the RC circuit R C
transient stable
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3-2 Nonlinear Models
??? separable variable ????????
3-2-1 Logistic Equation
used for describing the growth of population
The solution of a logistic equation is called the
logistic function.
Two stable conditions and
.
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Logistic curves for differential initial
conditions
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Solving the logistic equation
separable variable
?
(with initial condition P(0) P0)
logistic function
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Example 1 (text page 97) There are 1000
students. ? Suppose a student carrying a flu
virus returns to an isolate college campus of
1000 students. ? If it is assumed that the
rate at which the virus spreads is proportional
not only to the number x of infected students but
also to the number of students not infected,
?? ? x(0) 1
?? ?
k is a constant
? determine the number of infected students after
6 days
?? ? find x(6)
? if it is further observed that after 4 days
x(4) 50
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???????
Initial x(0) 1, x(4) 50
find x(6)
???separable variable ???
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Logistic equation ???
(1)
????????
????????????
(2)
????,???????
(3)
(4)
Gompertz DE
????? ?????,????
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3-2-2 ???????
A B ? C

• Use compounds A and B to for compound C
• x(t) the amount of C
• To form a unit of C requires s1 units of A and
  s2 units of B
• a the original amount of A
• b the original amount of B
• The rate of generating C is proportional to the
  product of the amount of A and the amount of B

See Example 2
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3-3 Modeling with Systems of DEs
Some Systems are hard to model by one dependent
variable but can be modeled by the 1st order
ordinary differential equation
They should be solved by the Laplace Transform
and other methods
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from Kirchhoffs 1st law
from Kirchhoffs 2nd law
(1)
(2)
Three dependent variable
We can only simplify it into two dependent
variable
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from Kirchhoffs 1st law
from Kirchhoffs 2nd law
(1)
(2)
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Chapter 3 ?????? variation ??????? DE ???
. the variation is proportional to
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??? Section 2-2 4, 6, 7, 9, 13, 14, 16, 21,
25, 28, 30, 36, 46, 48, 50, 54(a) Section 2-3
7, 9, 13, 15, 21, 29, 30, 33, 36, 40, 42, 53,
55(a), 56(a) Section 3-1 2, 5, 10, 15, 20,
29, 32 Section 3-2 2, 5, 14, 15 Section 3-3
12, 13 Review 3 3, 4, 11, 12