1 first-order differential equations s.-y. leu sept. 28, 2005
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First-Order Differential Equations
S.-Y. LeuSept. 28, 2005
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2.1 Solution Curves Without the Solution 2.2 Separable Variables2.3 Linear Equations2.4 Exact Equations2.5 Solutions by Substitutions2.6 A Numerical Solution2.7 Linear Models2.8 Nonlinear Models2.9 Systems: Linear and Nonlinear Models
CHAPTER 2First-Order Differential Equations
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Short tangent segments suggest the shape of the curve
2.1 Solution Curves Without the Solution
輪廓
Slope=
x
y
'ydx
dy
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The general first-order differential equation has the form
F(x, y, y’)=0or 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).
2.1 Solution Curves Without the Solution
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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.
2.1 Solution Curves Without the Solution
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Plotting Direction Fields 1st Step
y’=f(x,y)=C=constant curves of equal inclination
2nd StepAlong 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.
2.1 Solution Curves Without the Solution
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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.
2.1 Solution Curves Without the Solution
DEFINITION: autonomous DEA differential equation in which the independent variable does not explicitly appear is knownas an autonomous differential equation. For example, a first order autonomous DE hasthe form
DEFINITION: critical pointA critical point of an autonomous DE 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.
)(' yfy
)(' yfy
2.1 Solution Curves Without the Solution
DEFINITION: phase portraitA one dimensional phase portrait of an autonomous DE is a diagram which
indicatesthe 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.
)(' yfy
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DEFINITION: Separable DE
A first-order differential equation of the form
is said to be separable or to have separable variables.(Zill, Definition 2.1, page 44).
2.2 Separable Variables
)()( yhxgdx
dy
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Method of Solution
If represents a solution
2.2 Separable Variables
)()( yhxgdx
dy )()(
)(
1xg
dx
dyyp
dx
dy
yh
)()())(( ' xgxxp )(xy
dxxgdxxxp )()())(( '
21 )()( cxGcyH
dxxgdyyp )()( dxxdy )('
cxGxH )()(
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2.2 Separable Variables
The Natural Logarithm
domain , ,
The natural exponential function
domain
dtt
xx
1
1ln ),0(ln
xx elogln 71828.2e
xxDx
1ln
2
2 1ln
xxDx
baab lnln)ln( bab
alnlnln ara r lnln
xexexp),(exp
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DEFINITION: Linear EquationA first-order differential equation
of the form
is said to be a linear equation.(Zill, Definition 2.2, page 51).
When homogeneousOtherwise it is non-homogeneous.
2.3 Linear Equations
)()()( 01 xgyxadx
dyxa
0)( xg
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Standard Form
2.3 Linear Equations
)()( xfyxpdx
dy
)()()( 01 xgyxadx
dyxa
2.2 Separable Equations
A differential equation is called separable if it can be written as
Such that we can separate the variables and write
We attempt to integrate this equation
)()(' yBxAy
dxxAdyyB
)()(
1 0)( yB
dxxAdyyB
)()(
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2.2 Separable Equations
Example 1.is separable. Write
as Integrate this equation to obtain or in the explicit form What about y=0 ? Singular
solution !
xeyy 2'xey
dx
dy 2
dxey
dy x2 0y
key
x 1
key
x
1
2.2 Separable Equations Example 2.
is separable, too. We write
Integrate the separated equation to obtain
The general solution isAgain, 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
yyx 1'2
21 x
dx
y
dy
0x 1y
kx
y 1
1ln xxk Aeeey /1/11
xx BeAey /1/11
xBey /11
2.3 Linear Differential Equations
A first-order differential equation is linear if it has the form
Multiply the differential equation by to get
Now integrate to obtain
The function is called an integrating factor for the differential equation.
)()()(' xqyxpxy dxxpe )(
dxxpdxxpdxxp exqyexpxye )()(')( )()()(
dxxpdxxp exqexydx
d )()( )()(
Cdxexqexy dxxpdxxp )()( )()(
dxxpdxxpdxxp eCdxexqexy )()()( )()(
dxxpe )(
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 independent variables.
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.
)()()(' xqyxpxy
2.3 Linear Differential Equations Example is a linear DE. P(x)=1 and q
(x)=sin(x), both continuous for all x.An integrating factor is
Multiply the DE by to getOr
Integrate to get
The general solution is
)sin(' xyy
xdxdxxp eee )(
xe )sin(' xeyeey xxx )sin(
'xeye xx
Cxxedxxeyexxx )cos()sin(
2
1)sin(
xCexxy )cos()sin(2
1
2.3 Linear Differential Equations Example Solve the initial value problemIt can be written in linear form
An integrating factor is for Multiply the DE by to get OrIntegrate to get ,thenforFor the initial condition, we needC=17/4 the solution of the initial value
problem is
5)1(;3 2' yx
yxy
2' 31
xyx
y
xee xdxx )ln()/1( 0x
x 3' 3xyxy 3' 3xxy
Cxxy 4
4
3x
Cxy 3
4
3
0xCy
4
35)1(
xxxy
4
17
4
3)( 3