persamaan diferensial parsial dari persamaan linear dan kuasi orde pertama sampai materi persamaan Laplace.Deskripsi lengkap
persamaan diferensial parsial dari persamaan linear dan kuasi orde pertama sampai materi persamaan Laplace.Deskripsi lengkap
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Salah satu materi Perkuliahan Matematika III di Iniversitas Balikpapan yang membahas tentang langkah langkah penyelesaian Persamaan Diferensial Orde Satu dan Dua.Full description
Persamaan DIferensial dan AplikasiDeskripsi lengkap
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PEMECAHAN PERSAMAAN DIFERENSIAL ORDE PERTAMA
Untuk Untuk memeca memecahka hkan n suatu suatu persam persamaan aan diferen diferensia sial, l, kita kita harus harus mencari mencari fungsi yang memenuhi persamaan itu, yaitu dengan cara mengolah persamaan tersebu tersebutt sedemik sedemikian ian rupa rupa sehing sehingga ga semua semua koefisi koefisien en diferen diferensial sialnya nya hilang hilang dan tingga tinggallah llah hubung hubungan an antara antara y dan x. Pemeca Pemecahan han persam persamaan aan diferen diferensia siall orde orde pertama dapat dilakukan dengan menggunakan menggunakan : 1. Inte Integr gras asii Lang Langsu sung ng
Memberikan y = 2. Pemi Pemisa saha han n Vari Variab abel el F(y) .
Memberikan 3. Pers Persam amaa aan n Hom Homog ogen en Substitusikan y = vx Memberikan v + x 4. Pers Persam amaa aan n Line Linear ar Faktor Integrasi, FI = Dimana Memberika y FI = 5. Pers Persam amaa aan n Berno Bernoul ulli li
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A. Integrasi Integrasi Langsung Langsung Pemecahan persamaan diferensial dengan integrasi langsung dapat dilakukan
jika persamaannya dalam bentuk
Contoh : 1. Tentukan pemecahan umum dari persamaan
Penyelesaian : y = =
2. Tent Tentuk ukan an pem pemec ecaha ahan n umum umum dari dari persa persama maan an x
=
Penyelesaian :
y=
3. Tentukan pemecahan khusus dari persamaan
untuk x = 0 Penyelesaian :
, jika y = 3
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y= Jika y =3 untuk x = 0, maka : y=
B. Pemisahan Pemisahan variabel variabel Pemecahan persamaan diferensial diferensial dengan pemisahan variabel dapat dilakukan
jika persamaan yang diberikan berbentuk Contoh : 1. Tentukan pemecahan umum dari persamaan
Penyelesaian : (y + 1)
= 2x
2. Tentukan pemecahan umum dari persamaan
Penyelesaian :
(1+x) (1+y)
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3. Tentukan pemecahan umum dari persamaan
Penyelesaian :
ln (1+y) = ln (2+x) + C
4. Tentukan pemecahan umum dari persamaan
Penyelesaian :
dy =
dx
dx
ln y +
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Penyelesaian :
ln
= ln x + C
6. Tentukan pemecahan umum dari persamaan xy
Penyelesaian : y(y+1) dy =
dx
+ ln x + C
7. Tentukan pemecahan umum dari persamaan
Penyelesaian :
=
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ln y = ln x + x + C
8. Tentukan pemecahan umum dari persamaan
Penyelesaian :
C. Persamaan Persamaan Homogen Homogen – dengan subsitusi subsitusi y = vx Sifat persamaan diferensial homogen ditentukan oleh pangkat x dan y yang terli terliba batt dala dalam m masin masing-m g-masi asing ng suku suku deng dengan an dera deraja jatt yang yang sama. sama. Untu Untuk k memeca memecahka hkan n suatu suatu persam persamaan aan homoge homogen n adalah adalah menggu menggunak nakan an pemecah pemecahan an persamaan diferensial dengan substitusi y = vx. Bentuk ini dilakukan jika kita tidak dapat menyatakan ruas kanannya dalam bentuk ‘faktor x’ dan ‘faktor y’ (memecahkan persamaan dengan cara pemisahan variabel). Jadi substitusi y = vx merupakan suatu persamaan dimana v merupakan fungsi x. Contoh : 1. Tentukan pemecahan umum dari persamaan
Penyelesaian : Diferensiasikan terhadap x
=
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2 ln (1 + v) = ln x + C 2 ln (1 + ) = ln x + C
2. Tentukan pemecahan umum dari persamaan
Penyelesaian :
=
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( )2 = ln x + C
3. Tentukan pemecahan umum dari persamaan
Penyelesaian:
=
=
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2
}=
4. Tentukan pemecahan umum dari persamaan
Penyelesaian:
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+
= Q ; dimana dimana P dan Q adalah fungsi fungsi dari x atau kostanta. kostanta. Maka untuk untuk
memecahkan memecahkan persamaan semacam ini kita kalikan kalikan kedua ruasnya dengan faktor integrasi yang berbentuk
, FI =
{y
y y.FI = Dengan pengalian ini, ruas kirinya menjadi suatu koefisien diferensial lengkap. Perlu diingat bahwa dalam menetukan
, kita tidak menyertakan konstanta
integr integrasin asinya. ya. Didala Didalam m pemecah pemecahan an persama persamaan an difere diferensi nsial al yang yang menggu menggunak nakan an Faktor Integrasi (FI) ini, diperlukan hubungan logaritma. Misalkan jika : y =
maka y = F
Dimana F merupakan fungsi x Dan hal ini berarti juga bahwa :
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=
= x−3
Contoh : 1. Tentuk Tentukan an pemecah pemecahan an persam persamaan aan difere diferensi nsial al dari :
Penyelesaian :
FI =
=
y.FI =
y.
=
y.
=
y.
=
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Penyelesaian : FI =
y.FI =
y.ex =
y.ex =
y.ex = x + C y = xe-x + Ce-x
3. Tentuk Tentukan an pemecah pemecahan an persam persamaan aan difere diferensi nsial al dari :
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y.x-1 =
y.x-1 =
y.x-1 = x sin x –
y.x-1 = x sin x + cos x + C y = x2 sin x + x cos x + Cx
4. Tentuk Tentukan an pemecah pemecahan an persam persamaan aan difere diferensi nsial al dari : ( 1 + x2 )
Penyelesaian :
u =x
du = dx
dv = cos x
v = sin x
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y = x (1+x 2)-1/2 + C (1+x2)
5. Tentuk Tentukan an pemecah pemecahan an persam persamaan aan difere diferensi nsial al dari : ( 1 - x3 )
Penyelesaian :
( 1 - x3 ) : (1- x3)
y.FI =
FI =
=
=
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Jika ada suatu persamaan linear orde pertama yang berbentuk +
=Q
dengan dengan P dan Q adalah adalah fungsi fungsi x (atau (atau konstan konstanta), ta), maka maka langka langkah-. h-.
langkah yang dilakukan untuk memecahkan persamaan tersebut adalah sebagai berikut: (i)
Bagilah kedua ruasnya dengan y n , sehingga persamaannya menjadi
+P
(ii)
Misalkan z =
Sehingga dengan mendeferensiasikannya,
(iii)
Selanjutnya kalikan persamaan maka persamaannya menjadi
+P
dengan
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Contoh: 1. Tentukan pemecahan umum dari persamaan
Penyelesaian:
Mis:
FI =
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2. Tentukan pemecahan umum dari persamaan
Penyelesaian :
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3. Tentukan pemecahan umum dari persamaan
Penyelesaian :
Mis:
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