second order differential equation non...

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


Homogeneous & Non Homogenous –

Structure of the General Solution

0

0

)0(

)0(

yty

yty I.C.


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


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 )


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


Particular Solution For Non Homogeneous Equation

Class A


• 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



Class B

)(xgcyybya


• 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



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



Summary



Examples

teyyy 2343

0432

2

5

2

3

2

)4(493


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)(



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



tyyy sin243

Assume

tBtAtY

tBtAtY

tBtAtY

cossin)(

sincos)(

cossin)(



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



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)(



(Pathological Case) – Zill p.153



(Pathological Case) – Zill



(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)


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



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



- 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



)()()()()( 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



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


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



dttyyW

tgtytydt

tyyW

tgtytytYyp

)(,

)()()(

)(,

)()()()(

21

12

21

21

)()()()()( 2211 tytutytutYyp

dtt

eedt

t

eeececy

x

x

tx

x

x

txxx

002

1

2

121



pc yyy

- General Solution to the nonhomogeneous diff Eq.

second order differential equation non...

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