First-Order Differential Equations

1

First-Order Differential Equations

S.-Y. LeuSept. 28, 2005

2

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

3

Short tangent segments suggest the shape of the curve

2.1 Solution Curves Without the Solution

輪廓

Slope=

x

y

'ydx

dy

4

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

5

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

6

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

7

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

10

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

11

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 )()(

12

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

13

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

14

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

)()(

1

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