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Class Notes 5:
Second Order Differential Equation –
Non Homogeneous
82A – Engineering Mathematics
Second Order Linear Differential Equations –
Homogeneous & Non Homogenous v
• p, q, g are given, continuous functions on the open interval I
)(
0)()(
tgytqytpy
Homogeneous
Non-homogeneous
• Solution:
where
yc(x): solution of the homogeneous equation (complementary solution)
yp(x): any solution of the non-homogeneous equation (particular solution)
sHomogeneou
shomogeneou-Non
, 0
, )()()(
xgyxqyxpy
)()( xyxyy pc
Second Order Linear Differential Equations –
Homogeneous & Non Homogenous –
Structure of the General Solution
0
0
)0(
)0(
yty
yty I.C.
Second Order Linear Differential Equations –
Non Homogenous
)()()( tftqytpy
0
0
)0(
)0(
yty
yty I.C.
Theorem (3.5.1)
• If Y1 and Y2 are solutions of the nonhomogeneous equation
• Then Y1 - Y2 is a solution of the homogeneous equation
• If, in addition, {y1, y2} forms a fundamental solution set of the
homogeneous equation, then there exist constants c1 and c2 such
that
)()()()( 221121 tyctyctYtY
)()()( tgytqytpy
0)()( ytqytpy
Theorem (3.5.2) – General Solution
• The general solution of the nonhomogeneous equation
can be written in the form
where y1 and y2 form a fundamental solution set for the homogeneous
equation, c1 and c2 are arbitrary constants, and Y(t) is a specific
solution to the nonhomogeneous equation.
)()()()( 2211 tYtyctycty
)()()( tgytqytpy
• The methods of undetermined coefficients
• The methods of variation of parameters
Second Order Linear Non Homogenous Differential Equations –
Methods for Finding the Particular Solution
Make an initial assumption about the format of the particular solution Y(t) but with coefficients left unspecified
Substitute Y(t) into y’’+ p(t)y’+ q(t)y = g(t) and determine the coefficients to satisfy the equation
There is no solution of the form that we
assumed
Find a solution of
Y(t)
Determine the
coefficientsEnd
N Y
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Block Diagram
• Advantages
– Straight Forward Approach - It is a straight forward to execute
once the assumption is made regarding the form of the particular
solution Y(t)
• Disadvantages
– Constant Coefficients - Homogeneous equations with constant
coefficients
– Specific Nonhomogeneous Terms - Useful primarily for
equations for which we can easily write down the correct form of
the particular solution Y(t) in advanced for which the
Nonhomogenous term is restricted to
• Polynomic
• Exponential
• Trigonematirc (sin / cos )
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Block Diagram
• The particular solution yp for the nonhomogeneous equation
• Class A
)(xgcyybya
n
nn
n
axaxa
xPxg
...
)()(
1
10
xin Polynomial
0 ...
0,0 ...
0 ...
1
1
2
0
2
1
10
1
10
bcAxAxAx
bcAxAxAx
cAxAxA
y
n
n
n
nn
n
nn
p
Second Order Linear Non Homogenous Differential Equations –
Particular Solution For Non Homogeneous Equation
Class A
• The particular solution yp for the nonhomogeneous equation
• Class B
)...(
)()(
1
10 n
nnx
n
x
axaxae
xPexg
0)(
)...()(
1
10
ch
AxAxAexg n
nnx
equationstic characteri the of root a not is
0)(
)...()(
1
10
xch
AxAxAxexg n
nnx
equationstic characteri the of root simple a is
0)(
)...()(
1
10
2
xch
AxAxAexxg n
nnx
equationstic characteri the of root double a is
Second Order Linear Non Homogenous Differential Equations –
Particular Solution For Non Homogeneous Equation
Class B
)(xgcyybya
• The particular solution yp for the nonhomogeneous equation
• Class C
)...(
)(cos sin)(
1
10 n
nnxi
n
x
axaxae
xPxxexg
or
0)( ;...cos
...sin
0)( ;...cos
...sin
1
10
1
10
1
10
1
10
ichBxBxBx
AxAxAxxe
ichBxBxBx
AxAxAxe
y
n
nn
n
nn
x
n
nn
n
nn
x
p
)(xgcyybya
Second Order Linear Non Homogenous Differential Equations –
Particular Solution For Non Homogeneous Equation
Class C
• The particular solution of
s is the smallest non-negative integer (s=0, 1, or 2) that will ensure that no term in
Yi(t) is a solution of the corresponding homogeneous equation
s is the number of time
0 is the root of the characteristic equation
α is the root of the characteristic equation
α+iβ is the root of the characteristic equation
)(tgcyybya i
)(tg i )(tYi
n
nn
n atatatP ...)( 1
10 n
nns AtAtAt ...1
10
t
n etP )( t
n
nns eAtAtAt ...1
10
t
tetP t
n
cos
sin)(
teAtAtAt t
n
nns cos...1
10
teBtBtB t
n
nn sin...1
10
Second Order Linear Non Homogenous Differential Equations –
Particular Solution For Non Homogeneous Equation
Summary
Second Order Linear Non Homogenous Differential Equations –
Particular Solution For Non Homogeneous Equation
Examples
teyyy 2343
0432
2
5
2
3
2
)4(493
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 1
teyyy 2343
t
t
t
AetY
AetY
AetY
2
2
2
4)(
2)(
)(
tt
A
eeAAA 22
6
3464
2
1A
t
p etY 2
2
1)(
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 1
tAtY
tAtY
tAtY
sin)(
cos)(
sin)(
0cos3sin)52(
sin2sin4cos3sin
tAtA
ttAtAtA
Assume
There is no choice for constant A that makes the equation true for all t
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 2
tyyy sin243
Assume
tBtAtY
tBtAtY
tBtAtY
cossin)(
sincos)(
cossin)(
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 2
tyyy sin243
tttYp cos17
3sin17
5)(
053
235
BA
BA
ttBABtABA sin2cos)43(sin)43(
173
175 BA
teyyy t 2cos843
0102
8210
BA
BA
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 3
teBAteBAtY
teBAteBAtY
tBetAetY
tt
tt
tt
2sin)34(2cos)43()(
2sin)2(2cos)2()(
2sin2cos)(
132;1310 BA
tetetY tt
p 2sin13
22cos
13
10)(
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 4
(Pathological Case) – Zill p.153
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 4
(Pathological Case) – Zill p.153
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 5
(Pathological Case) – Zill
Second Order Linear Non Homogenous Differential Equations –
Method of Undermined Coefficients – Example 6
(Pathological Case) – Zillg(x)
Advantage – General method
Diff. eq. )()()( tgytqytpy
For the Homogeneous diff. eq.
0)()( ytqytpy
the general solution is
)()()( 2211 tyctyctyc
so far we solved it for homogeneous diff eq. with constant coefficients.
(Chapter 5 – non constant – series solution)
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters
Replace the constant by function21 &cc )(),( 21 tutu
)(
)(
22
11
tuc
tuc
)()()()((*) 2211 tytutytuy p
- Find such that is the solution to the nonhomogeneous diff. eq.
rather than the homogeneous eq.
)(),( 21 tutu
2222222211111111
22221111
yuyuyuyuyuyuyuyuy
yuyuyuyuy
p
p
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters
221122112222111122221111
2211
22221111
2222222211111111
])[(
)(
)()(
yuyuyuyupyuyuyuyuqyypyuqyypyu
yuyuxq
yuyuyuyuxp
yuyuyuyuyuyuyuyuyxqyxpy ppp
- Seek to determine 2 unknown function
- Impose a condition
- The two Eqs.
)(221122112211 tgyuyuyuyupyudx
dyu
dx
d
① ①
① ①
① ①
② ②⑤ ⑤ ③ ③
④ ④
= 0 = 0
①② ③ ④ ⑤
)(),( 21 tutu
0)()()()( 2211 tytutytu
)()()()()(
0)()()()(
2211
2211
tgtytutytu
tytutytu21
2121
,
,,,,
uu
yyyy
known
unknown
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters
- Seek to determine 2 unknown function
- Impose a condition Reducing the diff. equation to
- The two Eqs.
)(221122112211 tgyuyuyuyupyudx
dyu
dx
d
)(),( 21 tutu
0)()()()( 2211 tytutytu
)()()()()(
0)()()()(
2211
2211
tgtytutytu
tytutytu
21
2121
,
,,,,
uu
yyyy
known
unknown
)(221122112211 tgyuyuyuyupyuyudx
d
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters
)()()()()( 2211 tgtytutytu
21
21
1
1
2
21
21
2
2
1
0
;
0
yy
yy
gy
y
u
yy
yy
yg
y
u
),(;
),( 21
12
21
21
yyW
gyu
yyW
gyu
221
121
21
21
),(;
),(cdt
yyW
gyucdt
yyW
gyu
)()()()( 2211 tytutytuyp (*)onBased
221
121
21
21
),(),()( cdt
yyW
gyycdt
yyW
gyytYp
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters
Theorem (3.6.1)
• Consider the equations
• If the functions p, q and g are continuous on an open
interval I, and if y1 and y2 are fundamental solutions to Eq.
(2), then a particular solution of Eq. (1) is
and the general solution is
dttyyW
tgtytydt
tyyW
tgtytytY
)(,
)()()(
)(,
)()()()(
21
12
21
21
)()()()( 2211 tYtyctycty
)2(0)()(
)1()()()(
ytqytpy
tgytqytpy
2, 00
21
21
eeee
ee
yy
yyeeW
xx
xx
xx
xyy
1
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters – Example
xx
C ececy 21
1;101 21
2
- Solution to the homogeneous diff Eq.
- Solution to the nonhomogeneous diff Eq.
dtt
eu
xee
x
e
u
x
x
txx
x
0
2
1
2
)1(
2
10
11
dtt
eu
xexe
e
u
x
x
txx
x
0
2
1
2
)1(
2
10
22
,2
1
2
1)(
00
dtt
eedt
t
eetYy
x
x
tx
x
x
tx
p
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters – Example
dttyyW
tgtytydt
tyyW
tgtytytYyp
)(,
)()()(
)(,
)()()()(
21
12
21
21
)()()()()( 2211 tytutytutYyp
dtt
eedt
t
eeececy
x
x
tx
x
x
txxx
002
1
2
121
Second Order Linear Non Homogenous Differential Equations –
Method of Variation of Parameters – Example
pc yyy
- General Solution to the nonhomogeneous diff Eq.
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