minggu 2 1 engineering mathematics differential equations

8/9/2019 Minggu 2 1 Engineering Mathematics Differential Equations

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Dr. Ir. Harinaldi, M.EngMechanical Engineering Department

Faculty of Engineering University of Indonesia

DIFFERENTIALEQUATIONS


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i ff e ren t i a l Eq ua t ionsi ff e ren t i a l Eq ua t ions

An equation which involves unknown function and itsderivatives

ordinary differential equation (ode) : not involve partialderivativespartial differential equation (pde) : involves partial

derivativesorder of the differential equation is the order of thehighest derivatives

Examples:

second order ordinarydifferential equation

first order partial differentialequation

+ =2

23 sin

d y dy x y

dx dx ++ =

y y x t x

t x x t


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Modeling via Differential Equations

ote that the set of equations is called aModel for the system.How do we uild a Model!!he "asic steps in "uilding a model are:"tep #$ #learly state the assumptions on which the model will "e

"ased$ !hese assumptions should descri"e therelationships among the quantities to "e studied$

"tep %$#ompletely descri"e the parameters and varia"les to "e

used in the model$ "tep &$%se the assumptions (from &tep ') to derivemathematical equations relating the parameters andvaria"les (from &tep 2)$


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"ome 'pplication of Differential Equation in Engineering


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(a)u alir $ e liter*min

+ercampur sempurna

la)u alir $ f liter*min

'wal$ol. air $ o liter

Massa garam $a gram

-onsentrasi garam $ gram*liter

Massa garam setiap saat menit/ !

!

(arutan

'ir01aram


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A linear differential equation of order n is a differential equation written in the following form:

( ) ( ) ( ) ( )

+ + + + =K '

' '' ( )n n

n nn n

d y d y dy a x a x a x a x y f x

dx dx dx

where an(x) is not the ero function1eneral solution : looking for the unknown function of adifferential equation

2articular solution Initial alue 2ro lem/ : looking forthe unknown function of a differential equation where thevalues of the unknown function and its derivatives at somepoint are known

*ssues in finding solution :e3istence and uniqueness


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1 s tt O r d e r E - S e p a r a b le E q u a t io n sO r d e r E - S e p a r a b l e E q u a t io n s

!he differential equation M(x,y)dx + N(x,y)dy+ is separa"le ifthe equation can "e written in the form:

( ) ( ) ( ) ( )22'' =+ dy y g x f dx y g x f

&olution :

'$ ,ultiply the equation "y integrating factor: ( ) ( ) y g x f '2'

2$ !he varia"le are separated :( )( )

( )( )

('

2

2

' =+ dy y g y g

dx x f x f

3$ *ntegrating to find the solution:( )( )

( )( )

Cdy y g y g

dx x f x f =+

'

2

2

'


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Examples:

'$ &olve : dy x dx y dy x 2

- =Answer:


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Examples:

'$ &olve : y x

dxdy 22 +

=Answer:


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Examples:

2$ .ind the particular solution of : ( ) 2'/ '2 ==

y x

y

dx

dy

Answer:


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1 s tt O r d e r E - H o m o g e n e o u sO r d e r E - H o m o g e n e o u sEqua t i onsqua t ions0omogeneous .unction

f (x,y) is called homogenous of degreen if : ( ) ( ) y , x f y , x f n

=Examples:

( ) y x x y , x f 3- = homogeneous of degree -

( ) ( ) ( ) ( )( ) ( ) y x f y x x

y x x y x f 1

1-3--

3-

==

=

( ) y x x y x f cossin1 2 += non homogeneous( ) ( ) ( ) (

( ) ( )( ) y x f

y x x

y x x y x f

n 1

cossin

cossin122

2

+=+=


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1 s tt O r d e r E - H o m o g e n e o u sO r d e r E - H o m o g e n e o u sEqua t i onsqua t ions!he differential equationM(x,y)dx + N(x,y)dy+ is homogeneous

ifM(x,y)and N(x,y)are homogeneous and of the same degree

&olution :'$ %se the transformation to : dv x dx v dy vx y +==

2$ !he equation "ecome separa"le equation:( ) ( ) (11 =+ dv v x Qdx v x P

3$ %se solution method for separa"le equation

( )( ) ( )( )Cdv

v gv gdx

x f x f =+

'

2

2

'

-$ After integrating1v is replaced "y y/x


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1 s tt O r d e r E H o m o g e n e o u sO r d e r E H o m o g e n e o u sEqua t i onsqua t ionsExamples:

'$ &olve : ( )03 233 =+

dy xydx y xAnswer:


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1 s tt O r d e r E - H o m o g e n e o u sO r d e r E - H o m o g e n e o u sEqua t i onsqua t ionsExamples:

2$ &olve : 2

2

y x

y x

dx

dy

+

+=Answer:


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1 s tt O r d e r E E x a c t E q u a t io nO r d e r E E x a c t E q u a t io n

!he differential equationM(x,y)dx + N(x,y)dy+ is an exactequation if :

&olution :

!he solutions are given "y the implicit equation x N

y M

=

( ) C y x F =,

'$ *ntegrate eitherM(x,y) with respect to x or N(x,y)to y $Assume integratingM(x,y), then :

where : F/ x = M(x,y)and F/ y = N(x,y)

( ) ( ) ( )y dx y x M y x F += ,,

2$ ow : ( )[ ] ( ) ( )y x N y dx y x M y y

F ,', =+

=

or : ( ) ( ) ( )[ ]

= dx y x M y

y x N y ,,'


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3$ *ntegrate (y)to get (y) and write down the resultF(x,y) + C

Examples:'$ &olve : ( ) ( ) 01332 3 =+++ dy y x dx y x

Answer:


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Examples:

2$ &olve : ( )0cos214 2 =+++

dx

dy

y x xy Answer:


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1 s tt O r d e r E N o n E x a c t E q u a t io nO r d e r E N o n E x a c t E q u a t io n

!he differential equationM(x,y)dx + N(x,y)dy+ is a non exactequation if :

&olution :

!he solutions are given "y using integrating factor to change theequation into exact equation

x N

y M

'$ #heck if : ( ) only x of function x f N x N

y M

=

then integrating factor is( ) dx x f e

( ) only yof function y g M

y M

x N

= or if :

then integrating factor is( ) dy y g e


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2$ ,ultiply the differential equation with integrating factor whichresult an exact differential equation

3$ &olve the equation using procedure for an exact equation


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Examples:

'$ &olve : ( )022 =+++

dy xy dx x y x Answer:


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Examples:

2$ &olve : xy x y xy

dx

dy

+

+=2

23

Answer:


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1 s tt O rd e r E L in ea r Equ a t ionO rde r E L in ea r Eq u a t io n

Afirst order linear differential equation has the followinggeneral form:

( ) ( ) x qy x pdx dy =+&olution :

'$ .ind the integrating factor: ( ) ( ) = dx x pe x u

2$ Evaluate :

3$ .ind the solution:

( ) ( )( ) x u

C dx x q x uy +=

( ) ( )dx x q x u


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1 s tt O rd e r E L in ea r Equ a t ionO rde r E L in ea r Eq u a t io n

Examples:

'$ &olve : x xy

dx

dy 42 =+

Answer:


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2 ndd O rd e r E L i ne a r E q u a t ionO rd e r E L i n e a r Eq u a t ion

Asecond order differential equation is an equation involving the

unknown function y 1 its derivatives y 4 and y 441 and the varia"le x $5e will consider explicit differential equations of the form :

( ) x y y f dx

y d ,',2

2

=

Alinear second order differential equations is written as:

( ) ( ) ( ) ( ) x d y x c y x by x a =++

5hen d ( x ) + 1 the equation is calledhomogeneous 1 otherwise itis callednonhomogeneous


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!o a nonhomogeneous equation

( ) ( ) ( ) ( ) x d y x c y x by x a =++ we associate the so called associated homogeneous equation

( ) ( ) ( ) 0=++ y x c y x by x a

(NH)

(H)

Main result$ !he general solution to the equation ( 0) is given "y:

where: i/ y h is the general solution to the associated homogeneous

equation (H)/

ii/ y p is a particular solution to the equation (NH)$

ph y y y +=

d


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4asic property#onsider the homogeneous second order linear equation

( ) ( ) ( ) 0=++ y x c y x by x aor the explicit one

( ) ( ) 0=++ y x qy x py

2roperty$*f y 1 and y 2 are two solutions1 then:

( ) ( ) ( ) x y c x y c x y 2211 +=

is also a solution for any ar"itrary constantsc' 1c2

d


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2 ndd O r d e r E e d u c t io n o f O r d e rO r d e r E e d u c t io n o f O r d e r

!his technique is very important since it helps one to finda secondsolution independent froma 5nown one $

6eduction of 7rder +echnique

6et y ' "e a non ero solution of: ( ) ( ) 0=++ y x qy x py !hen1 a second solution y 2 independent of y ' can "e found as:

( ) ( ) ( ) x v x y x y 12 =5here:

( )( )

( ) = dx e x y x v dx x p

21

1

!he general solution is then given "y

( ) ( ) ( ) x y c x y c x y 2211 +=

dd


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2 ndd O r d e r E e d u c t io n o f O r d e rO r d e r E e d u c t io n o f O r d e r

Examples:'$ .ind the general solution to the 6egendre equation

( ) 0221 2 =+ y y x y x %sing the fact that : y 1 + x is a solution$

dd


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2 ndd O r d e r E H o m o g e n e o u s L E ! i t"O r d e r E H o m o g e n e o u s L E ! it "#o ns t an t #o e ff ic i en t so n s t an t # oe ff ic i en t sHomogeneous (inear Equations with 8onstant 8oefficients

A second order homogeneous equation with constant coefficientsis written as: ( )0 0 =++ acy y by a

where a1b and c are constant

!he steps to follow in order to find the general solution is as follows(1) W !t" d#$n th"characteristic equation

( )0 02 =++ ac ba

%h!& !& a ' ad at!c *"t 1 and 2 b" !t& ##t& $" hav"

aac bb

2

422,1

=

dd d


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2 ndd O r d e r E H o m o g e n e o u s L E ! i t"O r d e r E H o m o g e n e o u s L E ! it "#o ns t an t #o e ff ic i en t so n s t an t # oe ff ic i en t s

(2) f 1 and

2 a " d!&t!nct "a n -b" & (!fb2 - 4ac > 0 ), th"n th"

g"n" a t!#n !&. x x ec ec y 21 21 +=

(3) f 1 = 2 (!fb2 - 4ac = 0 ), th"n th" g"n" a t!#n !&. x x

xec ec y 11 21 +=(4) f 1 and 2 a " c#-p "x n -b" & (!fb2 - 4ac < 0 ), th"n th"

g"n" a t!#n !&.

( ) ( ) x ec x ec y x x

sincos 21 +=Wh" ".

abac

ab

2

4 and

2

2==

dd O d i


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2 ndd O r d e r E H o m o g e n e o u s L E ! i t"O r d e r E H o m o g e n e o u s L E ! it "#o ns t an t #o e ff ic i en t so n s t an t # oe ff ic i en t s

'$ .ind the solution to the *nitial 7alue 8ro"lem( ) ( ) 24 ; 24 ; 022 ===++ y y y y y

dd O d E N H L E


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2 ndd O r d e r E N o n H o m o g e n e o u s L EO r d e r E N o n H o m o g e n e o u s L E

!o a nonhomogeneous equation

( ) ( ) ( ) ( ) x d y x c y x by x a =++ we associate the so called associated homogeneous equation

( ) ( ) ( ) 0=++ y x c y x by x a

(NH)

(H)

Main result$ !he general solution to the equation ( 0) is given "y:

where: i/ y h is the general solution to the associated homogeneous

equation (H)/

ii/ y p is a particular solution to the equation (NH)$

ph y y y +=

ndd dO d E N H L E


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2 ndd O r d e r E N o n H o m o g e n e o u s L EO r d e r E N o n H o m o g e n e o u s L E$ e t " o d % n d e t e r m in e d # o e f fic ie n t se t " o d % n d e t e r m in e d # o e f f ic ie n t s5e will guess the form of y p and then plug it in the equation to find it$0owever1 it works only under the following two conditions:

the associated homogeneous equations has constant coefficientsthe nonhomogeneous termd ( x ) is a special form

( ) ( ) ( ) ( ) ( ) ( x e x L x d x e x P x d x

n

x

n sin or cos ==

ote: we may assume that d ( x ) is a sum of such functions where Pn( x ) and *n( x ) are polynomial functions of degreen

!hen a particular solution yp is given "y:

( ) ( ) ( ) ( ) ( )( ) x e x R x e x T x x y x n x ns p sincos +=( ) nnn x A x A x A A x T ++++= ...2210

( ) n

nn x B x B x BB x R ++++= ...2

210

5here:

2 ndd O d E N H L EO d E N H L E


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2 ndd O r d e r E N o n H o m o g e n e o u s L EO r d e r E N o n H o m o g e n e o u s L E$ e t " o d % n d e t e r m in e d # o e f fic ie n t se t " o d % n d e t e r m in e d # o e f f ic ie n t s!he steps to follow in applying this method:

'$ #heck that the two conditions are satisfied2$ 5rite down the characteristic equation and find its root

3$ 5rite down the num"er-$ #ompare this num"er to the roots of the characteristic equation

$ 5rite down the form of particular solution

9$ .ind constant and 0 "y plugging y p solution to original equation

02 =++ c ba i +

*f : 0 rootstheof onenotis =+ si 1 rootsdistinctheof oneis =+ si 2 rootsbothtoequalis =+ si

( ) ( ) ( ) ( ) ( )( ) x e x R x e x T x x y x n x ns p sincos +=( ) nnn x A x A x A A x T ++++= ...2210 ( )

nnn x B x B x BB x R ++++= ...

22109here$



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2 ndd O r d e r E N o n H o m o g e n e o u s L EO r d e r E N o n H o m o g e n e o u s L E$ e t " o d % n d e t e r m in e d # o e f fic ie n t se t " o d % n d e t e r m in e d # o e f f ic ie n t s

'$ .ind a particular solution to the equation( ) x y y y sin243 =



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*f the nonhomogeneous termd ( x ) consist of several terms:

( ) ( ) ( ) ( ) ( )=

==+++=

N i

i i N x d x d x d x d x d

121 ...

5e split the original equation intoN equations

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) x d y x c y x by x a

x d y x c y x by x a x d y x c y x by x a

N =++

=++ =++ 21

!hen find a particular solution y p!

A particular solution to the original equation is given "y:

( ) ( ) ( ) ( ) ( )=

=

=+++=N i

i pi pN p p p x y x y x y x y x y

121 ...

2 ndd O d E N H L EO d E N H g L E


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*f the nonhomogeneous termd ( x ) consist of several terms:

( ) ( ) ( ) ( ) ( )=

==+++=

N i

i i N x d x d x d x d x d

121 ...

5e split the original equation intoN equations

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) x d y x c y x by x a

x d y x c y x by x a x d y x c y x by x a

N =++

=++ =++ 21

!hen find a particular solution y p!

A particular solution to the original equation is given "y:

( ) ( ) ( ) ( ) ( )=

=

=+++=N i

i pi pN p p p x y x y x y x y x y

121 ...

2 ndd O d E N H L EO d E N H g L E


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'$ .ind a particular solution to the equation x x eey y y = 8343 2

minggu 2 1 engineering mathematics differential equations

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