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 xyp = − lnsec + tan cos +sin cos −sin cos
#
(
lnsecx+ tan x)
cos x −=
(
x x)
xx 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)
xx x
x x
yp lnsec3 tan3 sin 3
9 1 3 cos 3 tan 9 1 3 cos 3 1 + + − = #
(
x x)
xx 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)
xx 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
xx
+
x 2e
−4.
y” + 4 y’ + 4 y =
2x
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 =
2x
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