Persamaan Diferensial Orde II

PDB Orde II

PDB Orde II

″ ′

Solusi Homogen

Solusi Homogen

″ ′

!

"

# $

# $

% & '

≠

!

Solusi Homogen

Solusi Homogen

" & '

!

(_% (_"

$ &

! ) '

Contoh soal

Contoh soal

″

*

′

+

" $

," ,$

(_% (_"

″ + ′

-″ + ′

-$ $

,$

(_% (_"

″ .

.

r₁₂ = 2i 2

4 . 1 . 4

± = −

±

Persamaan Differensial non

Persamaan Differensial non

homogen

homogen

″ ′

≠

/

!

% !

Metode koefisien tak tentu

Metode koefisien tak tentu

r(x) yp

r(x) = emx yp = A emx

r(x) = Xn yp = AnXn + An-1Xn-1+…….+A1X + A0

r(x) = sin wx yp = A cos wx + B sin wx

r(x) =cos wx yp = A cos wx + B sin wx

r(x) = e uxsin wx yp = e ux (A cos wx + B sin wx )

R(x) =e uxcos wx yp = e ux (A cos wx + B sin wx )

( / )

#

Contoh

Contoh

% 1 2 $ 3 " # 4

" _{2 $ " 2 " 2 %}

/ _% " _" %

/ _% " _" %

# (_% " ₍

"

5 &

3 , & 1 &

6

& $ & " & + & & %7+

#

(_% " ₍

Contoh

Contoh

" 1 2 $ 3 " ) # 4

" _{2 $ " 2 " 2 %}

/ _% " _" %

# ( (

# (_% " ₍

"

5 & )

3 , & ) 1 , & ) 2

6

,& ) 2 2$ ,& ) " & ) )

,&,$ "& ) , $& " )

Contoh (no. 2 Lanjutan)

Contoh (no. 2 Lanjutan)

#

(_% " ₍

" %7% ) 2 $7%

& %7% ,$7%

$ 1 2 $ 3 " )

$ 1 2 $ 3 " )

# 4

) % "

(_% " ₍

Contoh

Contoh

. 1 2 $ 3 " % 3 ,%

# 4

" _{2 $ " 2 " 2 %}

/ _% " _" %

/ _% " _" %

# (_% " ₍

"

5 &

3 & & 1 "& & 6

"& &8 2 $ & & " & ,& & ,%

# (_% " ₍

Contoh

Contoh

(_% " ₍

" 2

6 % 3 ,%

3 "(_% " ₍

" 2 2

% (_% (_" "(_% (_"

(_% ,% (_" "

(_% ,% (_" "

#

Latihan

Latihan

$ . $ " - "

$ .

. "

$ .

. " " . " " " $ "

. . "

$ . $ " _"

- $ "

$

. . . !"

Metode Variasi Parameter

Metode Variasi Parameter

! ) ,

″

′

# ' # #

′

′

# #

′

#

Metode Variasi Parameter

Metode Variasi Parameter

′

′

#

′

″

′

′

″

#

′

′

#

″

/ ″ 4

′

′

″

#

′

′

#

″

" $

′

#

′

%

′

′

″

#

′

′

#

″

" $

′

#

′

%

$ # % $ %

$

″

"

′

% # $

″

"

′

%

′

′

#

′

′

$ %

Metode Variasi Parameter

Metode Variasi Parameter

Contoh

Contoh

% &

# 4

" _{% <}

# _' (_% ) (_"

# _' (_% ) (_"

5 #

)

3 , ₎

(

) " " _%

/

−

= x x dx

u 1 tan sin − = dx x x cos

sin2 −

− = dx x x cos cos 1 2 − −

Contoh (Lanjutan)

Contoh (Lanjutan)

x x

x tan sin

sec

ln + +

− = +

−

= secxdx cosxdx

= x x dx

v 1 tan cos ₌ dx x

sin = −cos x

/

# #

(

x x

)

x x x x x

y_p = − lnsec + tan cos +sin cos −sin cos

#

(

lnsecx+ tan x

)

cos x −

=

(

x x

)

x

x C

x C

Contoh

Contoh

" & ) # 4

" _{- < $}

# (_% ) $8 (_" $8

# (_% ) $8 (_" $8

5 _% 0 _"

% ) $ " $

3_% ,$ $ _{3" $ ) $}

(

$ ) " _$ " _$

/

−

= x x dx

u

3

3 sec 3

sin 2

−

= tan 3x dx

3

1 ₂

(

)

− −

= sec 3x 1 dx

3

Contoh (Lanjutan)

Contoh (Lanjutan)

x

x tan3

9 1 3 1 − = −

= dx sec 3xdx

3 1 3

1 2

= x x dx

v

3

3 sec 3

cos 2 _{= x dx}

3 sec 3

1

x x tan3 3 sec ln 9 1 + = / # #

(

x x

)

x

x x

x x

y_p lnsec3 tan3 sin 3

9 1 3 cos 3 tan 9 1 3 cos 3 1 + + − = #

(

x x

)

x

x x x C x C

y lnsec3 tan3 sin3

9 1 3 cos 3 1 3 sin 9 1 3 cos ₂

1 + − + + +

=

(

x x

)

x

x x

x lnsec3 tan3 sin 3

Latihan

Latihan

1.

y” + y = cosec x cot x

2.

y” + y = cot x

3.

y” – 3 y’ + 2y =

1

e

e

x

x

+

x 2

e

−

4.

y” + 4 y’ + 4 y =

₂

x

e

5.

y” + 4 y = 3 cosec 2x

6.

y” + 4 y = 3 cosec x

7.

4 y” + y = 2 sec (x/2)

8.

y” – 2y’ + y =

₂

x

x

1

Penerapan dalam Rangkaian Listrik

Penerapan dalam Rangkaian Listrik

) * =

+

0

,$-% 0 = 6 )

.

)

( )

-,

.

+

/-/.

)

/-.

/

*

_"

+

+

%

=

(Lanjutan)

(Lanjutan)

&

( )

-,

0

/0

)

0

/

*

%

>

"

=

+

+

/-/.

0

=

-( )

-,

0

+

/-/0

)

/-0

/

Contoh

Contoh

? @ A 4

BC( ) '1 *

+ , D

4

4 /

) BC(

%"

*

>

%+

E

"

.

+

.

+

.

=

+

"*

>

F

E

+

.

+

.

=

.

Contoh

Contoh

"* F

"

= +

+

$

.

±

−

=

2

/

(

+

-

+

-

)

.

_'

=

−. - _%

)

$

+

_"

$

/

(

+ - + -

)

. = " . % −$ + −. - _% ) $ + _" $

$

% .

" −

=

.

. 2

Rangkaian RLC

Rangkaian RLC

(

)

[

- -

]

. % −$ " . −. - " .) $ $ " $

4 . 0

$

"

$

"

%

−

−

=

+

$

% " . %

−

− =

+

#

(

)

[

- -

]

. =% −$ " . − −. - " .) $ + $ " $

-.

Latihan

Latihan

% =

BC( ) '1 *

+

,$-%

-4

" ? @ 4

B( )

+

Latihan

Latihan

$ = 0

BC( ) % *

$ * + " 8 % ,+

,$-% %" $GG

-4

. ? 0 4

-C( * % ,"

= + % ,G