02 ch 1 intro de (3).pptx
TRANSCRIPT
Introduction to Differential Equations
1.1 Definitions and Terminology
Definition:
A differential equation is an equation containing an unknown function and its derivatives.
32 xdx
dy
032
2
aydx
dy
dx
yd
364
3
3
ydx
dy
dx
yd
Examples:.
y is dependent variable and x is independent variable, and these are ordinary differential equations
1.
2.
3.
ordinary differential equations
Partial Differential Equation
Examples:
02
2
2
2
y
u
x
u
04
4
4
4
t
u
x
u
t
u
t
u
x
u
2
2
2
2
u is dependent variable and x and y are independent variables, and is partial differential equation.
u is dependent variable and x and t are independent variables
1.
2.
3.
Order of Differential Equation The order of the differential equation is order of the highest derivative in the differential equation.
Differential Equation ORDER
32 xdx
dy
0932
2
ydx
dy
dx
yd
364
3
3
ydx
dy
dx
yd
1
2
3
Degree of Differential Equation
Differential Equation Degree
032
2
aydx
dy
dx
yd
364
3
3
ydx
dy
dx
yd
0353
2
2
dx
dy
dx
yd
1
1
3
The degree of a differential equation is power of the highest order derivative term in the differential equation.
Linear Differential Equation A differential equation is linear, if
1. dependent variable and its derivatives are of degree one,2. coefficients of a term does not depend upon dependent
variable.
Example:
364
3
3
ydx
dy
dx
yd
is non - linear because in 2nd term is not of degree one.
.0932
2
ydx
dy
dx
ydExample:
is linear.
1.
2.
Example:3
2
22 x
dx
dyy
dx
ydx
is non - linear because in 2nd term coefficient depends on y.
3.
Example:
is non - linear because
ydx
dysin
!3
sin3y
yy is non – linear
4.
It is Ordinary/partial Differential equation of order… and of degree…, it is linear / non linear, with independent variable…, and dependent variable….
1st – order differential equation
2. Differential form:
01 ydxdyx.
.0),,( dx
dyyxf),( yxf
dx
dy
3. General form:
or
1. Derivative form:
xgyxadx
dyxa 01
10Differential Equation Chapter 1
First Order Ordinary Differential equation
11Differential Equation Chapter 1
Second order Ordinary Differential Equation
12Differential Equation Chapter 1
nth – order linear differential equation
1. nth – order linear differential equation with constant coefficients.
xgyadx
dya
dx
yda
dx
yda
dx
yda
n
n
nn
n
n
012
2
21
1
1 ....
2. nth – order linear differential equation with variable coefficients
xgyxadx
dyxa
dx
ydxa
dx
ydxa
dx
dyxa
n
n
nn
012
2
2
1
1 ......
13Differential Equation Chapter 1
Differential Equation Chapter 1 14
Solution of an Ordinary Differential Equation
Definition:
A solution of an nth-order ODE
is a function that possesses at least n derivatives and for which
for all x in I.
Interval of definition: You cannot think solution of an ODE without simultaneously thinking interval. The interval I is called interval of definition, the interval of existence, the interval of validity, or the domain of solution.
0),,',,( )( nyyyxF
0))(,),('),(,( )( xxxxF n
Differential Equation Chapter 1 15
Verification of a solution
Verify that the indicated function is a solution of the given DE on the interval (-∞, ∞).
a)
b)
,yxdx
dy 4
16
1xy
0'2'' yyy xxey
Differential Equation Chapter 1 16
Verification of an Implicit Solution
The relation is an implicit solution of DE
2522 yx
y
x
dx
dy
y=3x+c , is solution of the 1st order differential equation , c is arbitrary constant.
As y is a solution of the differential equation for every value of c, hence it is known as general solution.
3dx
dy
Examples
Examples
sin cosy x y x C
2 31 1 26 e 3 e ex x xy x y x C y x C x C
Observe that the set of solutions to the above 1st order equation has 1 parameter,
while the solutions to the above 2nd order equation depend on two parameters.
Families of Solutions 9 ' 4 0yy x
1 19 ' 4 9 ( ) '( ) 4yy x dx C y x y x dx xdx C
2 21This yields where .
4 9 18
Cy xC C
Example
Solution
The solution is a family of ellipses.
2
2 2 2 21 1 1
99 2 2 9 4 2
2
yydy x C x C y x C
Observe that given any point (x0,y0), there is a unique solution curve of the above equation which curve goes through the given point.
Diff
ere
ntia
l E
qua
tion
C
ha
pte
r 1
Families of Solutions
• A solution containing an arbitrary constant is called one-parameter family of solutions.
• A solution of a DE that is free of arbitrary parameters is called a particular solution.
19
Differential Equation Chapter 1 20
1.2 Initial-Value Problems1.2 Initial-Value Problems
21
Example
22
Interval I of Definition of a Solution
23
2
1y
x c
is a solution of the first-order differential equation 02 2' xyy
If we impose the initial condition y(0)=-1 we will get c = -1 and
therefore the solution will be 2
1.1
yx
We now emphasize the following three distinctions:
• Considered as a function, the domain of is R\{-1,1}.2
1
1y
x
24
• Considered as a solution of the differential equation the interval I of definition of could be taken to be any interval over
which y(x) is defined and differentiable. The largest intervals on which
is a solution are (-∞,-1), (-1,1) and (1,∞).
• Considered as a solution of the initial -value problem
y(0)=-1, the interval I of definition of could be taken to be any
interval over which y(x) is defined, differentiable, and contains the initial
point x = 0; the largest interval for which this is true is (-1,1).
02 2' xyy
2
1
1y
x
2
1
1y
x
,02 2' xyy
2
1
1y
x
25
Let R be a rectangle region in the xy-plane defined by
dycbxa ,
EXAMPLE
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Solution
27
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