Chapter 4: Higher-Order Differential Equations
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Transcript of Chapter 4: Higher-Order Differential Equations
Chapter 4: Higher-Order Differential Equations
Math-202
CH#1
Difinitions
CH#2
1st order DE
CH#4
Higher order DE
Chapter 4: Higher-Order Differential Equations
xyyexyy x cos7'''5'''3
xyyxyy cos7'''5'''3 2
xyyxyy coscos7'''5'''3
xyyxyy cos7'''5'''3
Sec 4.1: Linear DE (Basic Theory)
Sec 4.1.1: Initial Value Problem (IVP)
Boundary Value Problem (BVP)
)()(')()()( 01)1(
1)( xgyxayxayxayxa n
nn
n
,)( ,)(' ,)( 10)1(
1000 n
n yxyyxyyxy IVP: . nth order linear DE
Theroem 4.1 ( Existence of a Unique Solution)
Ixa
Ixgxaxaxaxa
Ix
n
nn
on 0)( )3
on cont are )( ),( ),(,),( ),( )2
)1
011
0
Sol y(x)
Exist
unique
:Example 0)1('' ,0)1(' ,0)1( ,cos7'''5'''3 yyyxyyexyy x1
Sec 4.1: Linear DE (Basic Theory)
Theroem 4.1 ( Existence of a Unique Solution)
Ixa
Ixgxaxaxaxa
Ix
n
nn
on 0)( )3
on cont are )( ),( ),(,),( ),( )2
)1
011
0
Sol y(x)
Exist
unique
:Example 0)1('' ,0)1(' ,0)1( ,cos7'''5'''3 yyyxyyexyy x1
10 40 124 )y'(,)y(x,yy''2 xeexy xx 33)( 22
10 30 62'22 )y'(,)y(,yxyy''x3 3)( 2 xxxy37)( 2 xxxy
Sec 4.1: Linear DE (Basic Theory)
Theroem 4.1 ( Existence of a Unique Solution)
Ixa
Ixgxaxaxaxa
Ix
n
nn
on 0)( )3
on cont are )( ),( ),(,),( ),( )2
)1
011
0
Sol y(x)
Exist
unique
:Example
10 00 3'')2( )y'(,)y(x,yyx9/p138
Find an interval centered about x=0 for which the given IVP has a unique solution
10 00 10)(tan'4
3'')25( 2
)y'(,)y(,xyxy
xyx2
Sec 4.1: Linear DE (Basic Theory)
50 40 124 )y'(,)y(x,yy''
Problem 1
51 40 124 )y(,)y(x,yy''
Problem 2
2ed order linear DE
What is the difference
IVP BVP
Sec 4.1: Linear DE (Basic Theory)
)()()()(
0
012
,yy(a)
,xgyxay'xay''xa
2ed order linear DE
IVP BVP
)()()()(
0
012
,yy(a)
,xgyxay'xay''xa
1' y(a)y 1yy(b)
Sec 4.1: Linear DE (Basic Theory)
10
012
)()()()(
yy'(a),yy(a)
,xgyxay'xay''xa
2ed order linear DE
IVP BVP10
012
)()()()(
yy(b),yy(a)
,xgyxay'xay''xa
Exist and unique
When??
BVP can have many, one, or No sol
0)8
(
0)0(
016''
y
y
yy
:Example
1)2
(
0)0(
016''
y
y
yy
0)2
(
0)0(
016''
y
y
yy
BVP3BVP2BVP1
xcxcxy 4sin4cos)( 21 2-parameter family of solutions Given that
0)(
0
0
2
1
xy
c
c
solutionno
c
01
01
xcxy
c
4sin)(
01
uniqueNo sol Infinity number of sol
Sec 4.1.2: Homogeneous Equations
0)(')()()( 01)1(
1)(
yxayxayxayxa nn
nn
)()(')()()( 01)1(
1)( xgyxayxayxayxa n
nn
n
diff
homogeneous nonhomogeneous
:Example
(*) 0cos7'''5'''3 xyyexyy x1
2 (**) 07'''5'''3 yyexyy x
(**) is the associated homogeneous DE of (*)
Remark: before we solve (*), we have to solve first (**)
Differential Operator
dx
dD
)( )1 2xD
:Example
)sin( )2 3 xxD
)( )3 22 xD
)3)(252( )4 2423 xxDDD
Differential Operators
operator polynomial
operator aldifferentiorder th -n
)()()()( 011
1 xaDxaDxaDxaL nn
nn
)3)(25( )5 24 xxxD
Properties: Differential Operator
aL[f(x)]L[af(x)] )1
L[g(x)]L[f(x)]g(x)]L[f(x) )2
)()()()( 011
1 xaDxaDxaDxaL nn
nn
bL[g(x)]aL[f(x)]bg(x)] L[af(x) )3
Linear Operator
Quiz on Monday
2.1
3.1
4.1.1
DE Differential Operator Form
:Example
356'5'' xyyy
xyyxxy sin6'5''' 2
gyL )(
gyL )(
:Example
33 xDDL xxg sinwhere
Write as DE
Homog DE
Theroem 4.2 ( Superposition Principle)
(*) of solutions twobe and Let 21 yy operatorlinear a is where L(*) 0L(y)sol also is c 1y
sol also is y21 y
:Example0423 yxy'y'''-x
21 xy
xxy ln22
are solutions1)Constant multiple is sol
2)Sum of two sol is also sol
3) Trivial sol is also a sol ??
Homog DE
Theroem 4.2 ( Superposition Principle)
(*) of solutions twobe ,,y,y,Let 321 kyy operatorlinear a is where L(*) 0L(y)
sol also is 2211 kk ycycyc
In general
Linear Dependence & Linear Independence
Definition 4.1
I(x), f(x),(x),ff n on dependent linearly functions 21 IF
such that zeros allnot , constantsexit there 21 nc,,cc
02211 (x) fc(x)fc(x)fc nn
(x) fc(x)fc(x)fc nn 2211Note: Linear Combination
:Example
32 21 x(x)f
122 x(x)f
32 (x)f 1
,6
,3
3
2
1
c
c
cIs this set linearly dependent ??
IF not then we say linearly independent
for every x in I
Linear Dependence & Linear Independence
Definition 4.1
I(x), f(x),(x),ff n on dependent linearly functions 21 IF
such that zeros allnot , constantsexit there 21 nc,,cc
02211 (x) fc(x)fc(x)fc nn
:Example
x(x)f 21 cos
x(x)f 22 sin
12 (x)f 1
,1
,1
3
2
1
c
c
cIs this set linearly dependent ??
IF not then we say linearly independent
for every x in I
Linear Dependence & Linear Independence
Definition 4.1
I(x), f(x),(x),ff n on dependent linearly functions 21 IF
such that zeros allnot , constantsexit there 21 nc,,cc
02211 (x) fc(x)fc(x)fc nnIF not then we say linearly independent
Special case dep. lin. are and 21 (x)f(x)fIf a set of two functions is lin. Dep, then one function is simply a constant multiple of the other.
:Examplex(x)f 2sin1
xx(x)f cossin2
Is this set linearly dependent ??
for every x in I
Linear Dependence & Linear Independence
Definition 4.1
I(x), f(x),(x),ff n on dependent linearly functions 21 IF
such that zeros allnot , constantsexit there 21 nc,,cc
02211 (x) fc(x)fc(x)fc nnIF not then we say linearly independent
:Examplex(x)f 1
x(x)f 2
Is this set linearly dependent ??
for every x in I
Linear Dependence & Linear Independence
Definition 4.1
I(x), f(x),(x),ff n on dependent linearly functions 21 IF
such that zeros allnot , constantsexit there 21 nc,,cc
02211 (x) fc(x)fc(x)fc nnIF not then we say linearly independent
Remark
A set of functions is linearly dependent if at least one function can be expressed as a linear combination of the remaining
:Example2
1 x(x)f x(x)f 2
Is this set linearly dependent ??
for every x in I
13 (x)f
532 24 xx(x)f
Homogeneous Equations
0)(')()()( 01)1(
1)(
yxayxayxayxa nn
nn
homogeneous
We are interested to find n linearly independent solutions
of the homog DE
)(,),(),(),( 321 xyxyxyxy n
Wronskian
Definition 4.2tdeterminan The s.derivative 1-n as Suppose 21 h(x), f(x),(x),ff n
called the Wronskian of the functions
)1()1(2
)1(1
21
21
21
'''),,,(
nn
nn
n
n
n
fff
fff
fff
fffW
:Example2
1 x(x)f x(x)f 2
Compute the Wroskian of these functions
23 x(x)f x(x)f 2
Compute the Wroskian of these functions
11 (x)f
Criterion for Linearly Independent Solutions
Theroem 4.3
:Examplexe(x)y 3
1 xe(x)y 32
These functions are solutions for the DE
0)(')()()( 01)1(
1)(
yxayxayxayxa nn
nn
THEN I,on DE homog for the solutions-n ,,, 21 nyyy
nyyy ,,, 21 Linearly Independent IxyyyW n in every for 0),,,( 21
09'' yy
lin. Indep ?
Fundamental set of solutions
:Example xe(x)y 31 xe(x)y 3
2
These functions are solutions for the DE 0'9''' yy
Fund. Set of sol. ?
Def 4.3 0)(')()()( 01)1(
1)(
yxayxayxayxa nn
nn
n are they ,,, )1 21 nyyy solutions ,,, )2 21 nyyy
indep. lin. ,,, )3 21 nyyy nyyy ,,, 21
Fundamental set of solutions
21, yy
xe(x)y 31 xe(x)y 3
2
These functions are solutions for the DE 0'9''' yy
Fund. Set of sol. ? 321 ,, yyy
13 (x)y
General Solution for Homog. DE
Theorem 4.5
THEN DE, theof solutions ofset fund. be Let 21 (x), y(x),(x),yy n
(x) yc(x)yc(x)ycy nn 2211
Is the general solution for the DE.
0)(')()()( 01)1(
1)(
yxayxayxayxa nn
nn
constants are where 21 n,c, ,cc
:Examplexe(x)y 3
1 xe(x)y 32
These functions are solutions for the DE 0'9''' yy
Find the general sol? 321 ,, yyy
13 (x)y
Given is a sol for 0'9''' yygeneral sol means what??
x(x)y 3sinh44
0)( yL
How to solve Homog. DE
Step 1
Step 2
Given a homg DE:
Find n-lin. Indep solutions
The general solution for the DE is
0)(')()()( 01)1(
1)(
yxayxayxayxa nn
nn
(x), y(x),(x),yy n21
(x) yc(x)yc(x)ycy nn 2211What is missing
0)( yL