Ch. 1 First-Order ODEs
Ch. 1 First-Order ODEs
Ordinary differential equations (ODEs)
• Deriving them from physical or other problems (modeling)
• Solving them by standard methods
• Interpreting solutions and their graphs in terms of a given problem
Ch. 1 First-Order ODEs 1.1 Basic Concepts. Modeling
1.1 Basic Concepts. Modeling
Ordinary Differential Equation : An equation that contains one or several derivatives of an
unknown function of one independent variable
Ex.
Partial Differential Equation
: An equation involving partial derivatives of an unknown function of two or more variables
Ex.
Differential Equation : An equation containing derivatives of an unknown function
Differential EquationOrdinary Differential Equation
Partial Differential Equation
2 2 2' cos , '' 9 0, ''' ' 2 '' 2xy x y y x y y e y x y
2 2
2 2 0u ux y
Ch. 1 First-Order ODEs 1.1 Basic Concepts. Modeling
Ex. (1) First order
(2) Second order
(3) Third order
Order : The highest derivative of the unknown function
2 2 2''' ' 2 '' 2 xx y y e y x y
First-order ODE : Equations contain only the first derivative and may contain y and any
given functions of x
, , ' 0F x y y
' ,y f x y• Explicit Form :
• Implicit Form :
'y
' cos y x
'' 9 0 y y
Ch. 1 First-Order ODEs 1.1 Basic Concepts. Modeling
Solution : Functions that make the equation hold true
Solution
• General Solution
: a solution containing an arbitrary constant
• Particular Solution
: a solution that we choose a specific constant
• Singular Solution
: an additional solution that cannot be obtained from the general solution
Ex.(Problem 16) ODE :
General solution :
Particular solution :
Singular solution :
2' ' 0y xy y
2y cx c
2 / 4y x
2 4y x
Ch. 1 First-Order ODEs
Initial Value Problems : An ordinary differential equation together with specified value
of the unknown function at a given point in the domain of the solution
0 0' , , y f x y y x y
Ex.4 Solve the initial value problem
Step 1 Find the general solution.
(see Example 3.) General solution :
Step 2 Apply the initial condition.
Particular solution :
7.50 ,3' yydxdyy
3xy x ce
7.50 0 ccey
xexy 37.5
1.1 Basic Concepts. Modeling
Ch. 1 First-Order ODEs
Modeling
The typical steps of modeling in detail
Step 1. The transition from the physical situation to its mathematical formulation
Step 2. The solution by a mathematical method
Step 3. The physical interpretation of differential equations and their applications
1.1 Basic Concepts. Modeling
Ch. 1 First-Order ODEs
Ex. 5 Given an amount of a radioactive substance, say 0.5 g(gram), find the amount present at any later time.
Physical Information.
Experiments show that at each instant a radioactive substance decays at a rate proportional to the
amount present.
Step 1 Setting up a mathematical model(a differential equation) of the physical process.
By the physical law :
The initial condition :
Step 2 Mathematical solution.
General solution :
Particular solution :
Always check your result :
Step 3 Interpretation of result.
The limit of as is zero.
dy dyy kydt dt
00.5 , 0 0.5 0.5ktdy ke ky y edt
0 0.5y
kty t ce
00 0.5 0.5 kty ce c y t e
y t
1.1 Basic Concepts. Modeling
Ch. 1 First-Order ODEs 1.2 Geometric Meaning of y` = f ( x , y ). Direction Fields
1.2 Geometric Meaning of y` = f ( x , y ). Direction Fields
Direction Field , y’=f(x,y) represents the slope of y(x)
For example, y’ = xy
- short straight line segments, lineal elements, can be drawn in xy-plane
- An approximate solution by connecting lineal elements, Fig.7(a)
Reason of importance of the direction field
• You do not have to solve the ODE to find y(x).
• The method shows the whole family of solutions and their typical properties., but its accuracy is limited
Ch. 1 First-Order ODEs
In this way, approximate sol is obtained. But it is sufficient.The exact solution can be obtained by the methods, in the following sections
Fig.7 CAS means computer algebra system (y(x)=1.213e^x^2/2)
Ch. 1 First-Order ODEs 1.3 Separable ODEs. Modeling
1.3 Separable ODEs. Modeling
Separable Equation :
Method of Separating Variables
dydx
dxdycdxxfdyxfyyg yg '
'g y y f x
A differential equation to be separable all the y ’s in the differential equation is on the one side and all the x ’s is on the differential equation is on the other side of the equal sign.
Ex. 1 Solve
2' 1y y
2 2 2
2
' /1 1 1 1 1
1 arctan tan 1
y dy dx dy dxy y y
dy dx c y x c y x cy
Ch. 1 First-Order ODEs 1.3 Separable ODEs. Modeling
Modeling Ex. 3 Mixing problems occur frequently in chemical industry. We explain here how to solve the basic model
involving a single tank. The tank in Fig.9 contains 1000gal of water in which initially 100lb of salt is dissolved.
Brine runs in at a rate of 10gal/min, and each gallon contains 5lb of dissolved salt. The mixture in the tank is
kept uniform by stirring. Brine runs out at 10 gal/min. Find the amount of salt in the tank at any time t.
Step 1 Setting up a model.
▶ Salt’s time rate of change = Salt inflow rate – Salt outflow rate “Balance law”
Salt inflow rate = 10 gal/min × 5 lb/gal = 50 lb/min
Salt outflow rate = 10 gal/min × y/1000 lb/gal = y/100 lb/min
▶ The initial condition :
Step 2 Solution of the model.
▶ General solution :
▶ Particular solution :
'/ ydtdy
yyy 5000100
1100
50'
1000 y
1001 1 ln 5000 * 50005000 100 100
tdy dt y t c y cey
1000 49005000 4900 100500050000t
eycccey
Ch. 1 First-Order ODEs
Extended Method : Reduction to Separable Form
Certain first order equations that are not separable can be made separable by a simple
change of variables.
▶ A homogeneous ODE can be reduced to separable form by the substitution of y=ux
Ex. 6 Solve
' yy fx
' ' & ' ' 'y du dx yy f u x u f u y ux u y ux u x ux f u u x x
22'2 xyxyy
2 22
22 2 2
1 1 1 2 12 ' ' ' 2 2 1
1 1
y x uxyy y x y u x u u du dxx y u u x
c y cu x y cxx x x
1.3 Separable ODEs. Modeling
Ch. 1 First-Order ODEs 1.4 Exact ODEs, Integrating Factors
1.4 Exact ODEs, Integrating Factors
Exact Differential Equation : The ODE M(x , y)dx +N(x , y)dy =0 whose the differential
form
M(x , y)dx +N(x , y)dy is exact, that is, this form is the differential .
If ODE is an exact differential equation, then
Condition for exactness :
Solve the exact differential equation
• get
• get
, , 0 0 ,M x y dx N x y dy du u x y c
u udu dx dyx y
2
M N M u u u Ny x y y x x y x y x
, , , , u uM x y u x y M x y dx k y N x yx y
, , , , u uN x y u x y N x y dy l x M x yy x
& dk k ydy
& dl l xdx
Ch. 1 First-Order ODEs
Ex. 1 Solve
Step 1 Test for exactness.
Step 2 Implicit general solution.
Step 3 Checking an implicit solution.
2cos 3 2 cos 0x y dx y y x y dy
, cos sinMM x y x y x yy
2, 3 2 cos sinNN x y y y x y x yx
M Ny x
2 3 2
, , cos sin
cos , 3 2 *
u x y M x y dx k y x y dx k y x y k y
u dk dkx y N x y y y k y y cy dy dy
3 2 , sinu x y x y y y c
2 2
2
cos cos ' 3 ' 2 ' 0 cos cos 3 2 ' 0
cos 3 2 cos 0
u x y x y y y y yy x y x y y y yx
x y dx y y x y dy
1.4 Exact ODEs, Integrating Factors
Ch. 1 First-Order ODEs 1.4 Exact ODEs, Integrating Factors
Reduction to Exact Form, Integrating Factors
Some equations can be made exact by multiplication by some function, ,
which is usually called the Integrating Factor.
, 0F x y
Ex. 3 Consider the equation
That equation is not exact.
If we multiply it by , we get an exact equation
0ydx xdy
1, 1 y xy x
2 2 2
1 1 10 y ydx dyx x y x x x x
2
1x
Ch. 1 First-Order ODEs
How to Find Integrating Factors
The exactness condition :
Golden Rule : If you cannot solve your problem, try to solve a simpler one.
Hence we look for an integrating factor depending only on one variable.
Case 1)
Case 2)
0FPdx FQdy
F P F QFP FQ P F P Fy x y y x x
0F FF F x F', x y
1 1' where
exp
y xdF P QFP F Q FQ R x R x
F dx Q y x
F x R x dx
1 * 1* * * where * * exp **
dF Q PF F y R R F y R y dyF dx P x y
1.4 Exact ODEs, Integrating Factors
Ch. 1 First-Order ODEs
Ex. Find an integrating factor and solve the initial value problem
Step 1 Nonexactness.
Step 2 Integrating factor. General solution.
The general solution is
Step 3 Particular solution
1 0, 0 1x y y ye ye dx xe dy y
yyyxyyx yeeeyPyeeyxP
,
yy exQxeyxQ
1,xQ
yP
1 1 1 1 1
x y y y y x y yy y
P QR e e ye e e yeQ y x xe xe
Fails.
1 1* 1 *y x y y y yx y y
Q PR e e e ye F y eP x y e ye
0x ye y dx x e dy is the exact equation.
' ' , x x y y yuu e y dx e xy k y x k y x e k y e k y ey
cexyeyxu yx ,
00 1 0, 1 0 3.72 , 3.72x yy u e e u x y e xy e
1.4 Exact ODEs, Integrating Factors
Ch. 1 First-Order ODEs 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics
1.5 Linear ODEs. Bernoulli Equation. Population Dynamics
ODEs
Linear ODEs
Nonlinear ODEs
Homogeneous Linear ODEs
Nonhomogeneous Linear ODEs
Linear ODEs : ODEs which is linear in both the unknown function and its derivative.
Ex. : Linear differential equation
: Nonlinear differential equation
• Standard Form : ( r(x) : Input, y(x) : Output )
Homogeneous, Nonhomogeneous Linear ODE
: Homogeneous Linear ODE
: Nonhomogeneous Linear ODE
'y p x y r x
' 0y p x y r x
2'y p x y r x y
'y p x y r x
' 0y p x y
Ch. 1 First-Order ODEs 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics
Homogeneous Linear ODE.(Apply the method of separating variables)
Nonhomogeneous Linear ODE.(Find integrating factor and solve )
is not exact
• Find integrating factor.
• Solve
' 0 ln * p x dxdyy p x y p x dx y p x dx c y ce
y
' 0y p x y r x py r dx dy 0 1py r py x
1 1 pdxP Q dFR p p F e
Q y x F dx
y 0pdx pdx
e py r dx e d
Ex. 1 Solve the linear ODE xeyy 2'
2 2 21, , x h h x x x x x x xp r e h pdx x y e e rdx c e e e dx c e e c e ce
' ' ,
pdx pdx pdx pdx pdx
pdx pdx pdx pdx pdx pdx
uu ye l x pye l x e py r l x re l x re dx cx
u ye re dx c ye re dx c y e re dx c
Ch. 1 First-Order ODEs 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics
Bernoulli Equation :
We set
' 0 & 1 ay p x y g x y a
1 ' 1 ' 1 1 1
' 1 1
a a a au a y y a y gy py a g py a g pu
u a pu a g
: the linear ODE
1 au x y x
Ex. 4 Logistic Equation
Solve the following Bernoulli equation, known as the logistic equation (or Verhulst equation)
The general solution of the Verhulst equation is
2' ByAyy
2 2 1
2 2 2 1
' ' & 2
' ' '
, & h h Ax Ax
y Ay By y Ay By a u y
u y y y Ay By Ay B Au B u Au B
Bp A r B h pdx Ax u e e rdx c e eA
Ax Bc ceA
1 1
Axy
Bu ceA
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