ࡱ>  Okq` 0'YbjbjqPqP ?::PI@@@@   4...80/44KD5U"2U2U2UqVqVqV7hh 5\mVqV5\5\@@2U2U;5\@2U 2U5\:@ 2U85 `DB.uV"0Kx}D 5\5\5\5\qVqZ,[^5\5\5\K5\5\5\5\444).444.444@@@@@@ BAB III PERSAMAAN DIFFERENSIAL LINEAR 3.1 Bentuk Umum Persamaan differensial linear dikategorikan sebagai persamaan differensial tingkat satu derajat satu, sehingga bentuk umum persamaan differensial linear dapat dinyatakan dengan M(x,y) dx + N(x,y) dy = 0. Dalam hal yang lebih khusus persamaan differensial linear tingkat satu dinyatakan dalam bentuk umum p1(x)  EMBED Equation.3  + po(x) y = q(x) dimana p1(x)  EMBED Equation.3  0, po(x), q(x) adalah fungsi x yang tidak bergantung kepada y. Jika masing-masing bagian pada persamaan differensial linear di atas diballgi dengan p1(x) maka diperoleh bentuk:  EMBED Equation.3 +  EMBED Equation.3  EMBED Equation.3  y =  EMBED Equation.3   EMBED Equation.3  + P(x) y = Q(x), dimana P(x) =  EMBED Equation.3  dan Q(x) =  EMBED Equation.3  P(x) dan Q(x) kontinu dalam suatu interval I  EMBED Equation.3  Real. Contoh  EMBED Equation.3  2xy = 4x, P(x) = 2x, Q(x) = 4x  y = (2+2x), P(x) = 1, Q(x) = (2+2x) x EMBED Equation.3  (2-3x EMBED Equation.3 )y = x EMBED Equation.3   EMBED Equation.3 , P(x) =  EMBED Equation.3 , Q(x) = 1 y ln y dx + (x-ln y) dy = 0  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3 , P(y) =  EMBED Equation.3 , Q(y) =  EMBED Equation.3  sin x  (cos x) y = x2 sin x  EMBED Equation.3  EMBED Equation.3 , P(x) = cot x, Q(x) = x EMBED Equation.3  Contoh persamaan 1-5 di atas adalah persamaan differensial linear (tingkat 1). Selanjutnya perhatikan persamaan differensial tingkat satu derajat satu di bawah ini. 6.  3xy2 = sin x, karena ada y2  (sin x) y3 = ex + 1, karena ada  EMBED Equation.3   EMBED Equation.3  - y = xy EMBED Equation.3 , karena Q(x) tergantung selain x, yaitu y EMBED Equation.3   EMBED Equation.3  + y = x2y, karena Q(x) tergantung selain x, yaitu y. x dy + y EMBED Equation.3 x dx = 0. Contoh 6-10 di atas bukan persamaan differensial linear karena Q(x) bukan fungsi x yang tidak bergantung pada y sebagaimana syarat yang disebutkan dalam bentuk umum dan tidak sesuai dengan bentuk  + P(x) y = Q(x). 3.2 Cara Menentukan Selesaian Persamaan Liner Persamaan differensial linear  + P(x) y = Q(x) dapat ditentukan selesaian umumnya dengan beberapa cara. Masing-masing cara menggunakan pendekatan yang berbeda, walaupun pada akhirnya diperoleh bentuk umum selesaian yang sama. Cara tersebut adalah: 1) faktor integral, 2) metode Lagrange, 3) mengubah menjadi PD eksak, dan 4) persamaan Bernoulli. Cara Faktor Intregral Misal selesaian  + P(x) y = Q(x) adalah y = uv, dimana u dan v masing-masing fungsi dari x sehingga y = uv + uv. Dengan mensubstitusikan y dan y ke persamaan  + P(x) y = Q(x).  EMBED Equation.3 (uv + uv) + P(x) uv = Q(x)  EMBED Equation.3 v(u + P(x)u) + uv = Q(x) Jika dimisalkan u + P(x)u = 0 maka uv = Q(x), Akibatnya untuk u + P(x) u = 0 diperoleh  EMBED Equation.3  = - P(x)u  EMBED Equation.3  EMBED Equation.3  = -P(x) dx  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  = -  EMBED Equation.3 P(x) dx  EMBED Equation.3 Ln u = -  EMBED Equation.3 P(x) dx  EMBED Equation.3 u =  EMBED Equation.3  Substitusikan u ke uv = Q(x), sehingga didapatkan v =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  dv =  EMBED Equation.3   EMBED Equation.3 v =  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  + C =  EMBED Equation.3  EMBED Equation.3  v =  EMBED Equation.3 Q(x)  EMBED Equation.3  dx + C Karena selesaiannya y = uv, maka selesaian umum (primitif) persamaan differensial linear  + P(x) y = Q(x) adalah y =  EMBED Equation.3  ( EMBED Equation.3 Q(x)  EMBED Equation.3  dx + C )  EMBED Equation.3  y  EMBED Equation.3  =  EMBED Equation.3 Q(x)  EMBED Equation.3  dx + C selanjutnya  EMBED Equation.3  dinamakan faktor integral (I) Contoh soal  EMBED Equation.3 Tentukan selesaian umum persamaan differensial liner di bawah ini  y = (2+2x) Jawab P(x) = 1 dan Q(x) = (2+2x) Faktor integralnya I =  EMBED Equation.3  = e EMBED Equation.3  Sehingga selesaian umumnya persamaan  y = (2+2x) adalah ye EMBED Equation.3  =  EMBED Equation.3 ) e EMBED Equation.3 dx y = EMBED Equation.3  EMBED Equation.3   EMBED Equation.3 e dx + e EMBED Equation.3   EMBED Equation.3 2ex dx =  EMBED Equation.3  ( 2e EMBED Equation.3 +C ) + 2e EMBED Equation.3  (xe EMBED Equation.3 - e EMBED Equation.3 +C) = 2 + Ce EMBED Equation.3 + 2x 2 + Ce EMBED Equation.3  = 2x + 2ce EMBED Equation.3  = 2x + ce EMBED Equation.3  y ln y dx + (x-ln y) dy = 0 Jawab y ln y dx + (x-ln y) dy = 0  EMBED Equation.3  +  EMBED Equation.3 x =  EMBED Equation.3  P(y) =  EMBED Equation.3  dan Q(y) =  EMBED Equation.3  Faktor integral e EMBED Equation.3  = e EMBED Equation.3  = e EMBED Equation.3 = Ln y Selesaian umumnya xe EMBED Equation.3  =  EMBED Equation.3  dy x Ln y =  EMBED Equation.3 y dy =  EMBED Equation.3 y d(ln y) = ln EMBED Equation.3  + c Persamaan linear y ln y dx + (x-lny) dy = 0 mempunyai selesaian umum 2x Ln y = Ln EMBED Equation.3  + c EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  x EMBED Equation.3  (2-3x EMBED Equation.3 ) y = x EMBED Equation.3  Jawab Persamaan di atas dibagi dengan x EMBED Equation.3  diperoleh persamaan linear baru  EMBED Equation.3  + ( EMBED Equation.3 ) y = 1 P(x) = ( EMBED Equation.3 ) dan Q(y) = 1 Sehingga faktor integralnya e = e EMBED Equation.3  = e EMBED Equation.3  Selesaian umumnya ye EMBED Equation.3  =  EMBED Equation.3 dx =  EMBED Equation.3  EMBED Equation.3 1.e EMBED Equation.3 dx =  EMBED Equation.3 e EMBED Equation.3 e EMBED Equation.3  dx y e EMBED Equation.3 = (1/2 e  EMBED Equation.3  + c) Persamaan differensial linear x EMBED Equation.3  (2-3x EMBED Equation.3 ) y = x EMBED Equation.3  mempunyai selesaian y e EMBED Equation.3 = (1/2 e  EMBED Equation.3  + c) Cara LAGRANGE Menyelesaikan persamaan differensial linear  + P(x) y = Q(x) dapat juga dilakukan dengan Cara Lagrange. Cara ini dilakukan dengan mengubah persamaan linear sehingga ruas kanan sama dengan 0 dan mengubah konstanta C menjadi fungsi dari x atau C(x). Perhatikan kembali persamaan  + P(x) y = Q(x)  EMBED Equation.3  y + P(x)y = Q(x) Ambil y + P(x)y = 0, maka  EMBED Equation.3  EMBED Equation.3 = -P(x)y  EMBED Equation.3  EMBED Equation.3  = -P(x) dx  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3 -P(x) dx  EMBED Equation.3  ln y =  EMBED Equation.3 -P(x) dx  EMBED Equation.3  y = e EMBED Equation.3   EMBED Equation.3  y = e EMBED Equation.3   EMBED Equation.3  y = e EMBED Equation.3 .e EMBED Equation.3   EMBED Equation.3  y = c(x) e EMBED Equation.3  Selanjutnya akan dicari fungsi c(x) dari persamaan  EMBED Equation.3 y = c(x) e EMBED Equation.3 , maka Ln y = ln (c(x) e EMBED Equation.3 )  EMBED Equation.3  ln y = ln c(x) + Ln e EMBED Equation.3   EMBED Equation.3  ln y = ln c(x) -  EMBED Equation.3  Jika persamaan di atas didefferensialkan terhadap x, diperoleh:  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 y ( EMBED Equation.3 )  EMBED Equation.3   EMBED Equation.3 =  EMBED Equation.3  Dari persamaan y = c(x) e EMBED Equation.3  diperoleh:  EMBED Equation.3  P(x) y =  EMBED Equation.3   EMBED Equation.3  Q(x) = e EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 = Q(x) e EMBED Equation.3   EMBED Equation.3 c(x) =  EMBED Equation.3  Q(x) e EMBED Equation.3  dx Dengan mensubtitusikannya ke dalam y = c(x) e EMBED Equation.3  maka diperoleh selesaian umum persamaan dengan metode Lagrange y = e EMBED Equation.3  ( EMBED Equation.3  Q(x) e EMBED Equation.3  dx ) Contoh soal Tentukan selesaian umum persamaan 1.  EMBED Equation.3 + y Cotgn x = 5e EMBED Equation.3  EMBED Equation.3  Jawab P(x) = cotgn x dan Q(x) = 5ecos x Sehingga faktor integralnya e EMBED Equation.3 = eln sin x = sin x. Selesaian umum persamaan yang dicari adalah: ye EMBED Equation.3  =  EMBED Equation.3  dx  EMBED Equation.3 y sin x =  EMBED Equation.3 sin x dx  EMBED Equation.3  y sin x =  EMBED Equation.3 d(-cos x)  EMBED Equation.3 y sin x = 5(-e EMBED Equation.3 ) + C (x-2)  EMBED Equation.3  = y + 2(x-2) EMBED Equation.3  Jawab Persamaan dibagi dengan (x-2) diperoleh:  EMBED Equation.3  -  EMBED Equation.3  = 2(x-2) EMBED Equation.3  P(x) =  EMBED Equation.3  dan Q(x) = 2(x-2) EMBED Equation.3  Faktor integral e EMBED Equation.3  = e EMBED Equation.3  = e EMBED Equation.3  =  EMBED Equation.3  Selesaian umum persamaan diperoleh  EMBED Equation.3 + y Cotgn x = 5e EMBED Equation.3  ye EMBED Equation.3  =  EMBED Equation.3  dx  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3  dx  EMBED Equation.3  EMBED Equation.3  = 2  EMBED Equation.3 dx  EMBED Equation.3  Cara Mengubah menjadi Persamaan Differensial Eksak. Karena  + P(x) y = Q(x) atau (P(x)y Q(x)) dx + dy = 0 belum merupakan persamaan differensial eksak untuk P(x)  EMBED Equation.3 0, maka perlu mencari faktor integralnya. Misal u(x) faktor integral, maka u(x)[ (P(x)y Q(x)) dx + dy = 0 ]  EMBED Equation.3 [u(x)P(x)y u(x)Q(x)] dx + u(x) dy = 0 merupakan persamaan differensial eksak. Berdasarkan syarat persamaan differensial eksak diperoleh  EMBED Equation.3  = u(x)P(x) dan  EMBED Equation.3 =  EMBED Equation.3  sehingga  EMBED Equation.3  = u(x)P(x)  EMBED Equation.3 = P(x) dx  EMBED Equation.3   EMBED Equation.3 = P(x) dx  EMBED Equation.3  ln u(x) =  EMBED Equation.3 P(x) dx u(x) = e EMBED Equation.3  Selanjutnya jika nilai u(x) dikalikan dengan  + P(x) y = Q(x), diperoleh u(x)  + u(x) P(x) y = u(x) Q(x) Karena  EMBED Equation.3 = u(x) P(x), maka u(x)  +  EMBED Equation.3 = u(x) Q(x)  EMBED Equation.3  EMBED Equation.3 (u(x)y) = u(x)Q(x) Dengan mengintegralkan persamaan terakhir terhadap x diperoleh  EMBED Equation.3  EMBED Equation.3 (u(x)y) =  EMBED Equation.3 u(x)Q(x) dx  EMBED Equation.3 u(x) y =  EMBED Equation.3   EMBED Equation.3 y = u(x) EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 y = e EMBED Equation.3  adalah selesaian umumnya. Karena u(x) = e EMBED Equation.3  maka diperoleh y = e EMBED Equation.3  EMBED Equation.3 e EMBED Equation.3 dx Contoh Tentukan selesaian umum persamaan 1.  EMBED Equation.3 -  EMBED Equation.3  = x cos x Jawab Kalikan persamaan dengan x, sehingga didapat  EMBED Equation.3  -  EMBED Equation.3 = x EMBED Equation.3 cos x P(x) =  EMBED Equation.3  dan Q(x) = x EMBED Equation.3 cos x u(x) = e EMBED Equation.3  = e EMBED Equation.3  =  EMBED Equation.3  Jika persamaan  EMBED Equation.3  -  EMBED Equation.3 = x EMBED Equation.3 cos x dikalikan dengan u(x) =  EMBED Equation.3  diperoleh x EMBED Equation.3   EMBED Equation.3  - 2yx EMBED Equation.3  = cos x  EMBED Equation.3  EMBED Equation.3  = cos x  EMBED Equation.3  x EMBED Equation.3 y =  EMBED Equation.3 x dx  EMBED Equation.3  y = x EMBED Equation.3 sin x + cx EMBED Equation.3  EMBED Equation.3  2. y EMBED Equation.3 dx + (3xy-1) dy = 0 Jawab y EMBED Equation.3 dx + (3xy-1) dy = 0  EMBED Equation.3  EMBED Equation.3 +  EMBED Equation.3  =  EMBED Equation.3  P(y) =  EMBED Equation.3  Q(y) =  EMBED Equation.3  u(x) = e EMBED Equation.3  = y EMBED Equation.3  Selesaian umum persamaan di atas xe EMBED Equation.3  =  EMBED Equation.3 dy xy EMBED Equation.3  =  EMBED Equation.3 y EMBED Equation.3  dy xy EMBED Equation.3  = y EMBED Equation.3  + c EMBED Equation.3  Persamaan BERNOULLI Persamaan differensial linear disebut persamaan Bernoulli jika bentuk umumnya  + P(x) y = y EMBED Equation.3 Q(x),  EMBED Equation.3 y EMBED Equation.3  + P(x) y EMBED Equation.3  = Q(x) Untuk menentukan selesaian umumnya misalkan y EMBED Equation.3 = v. Dengan menurunkan terhadap variabel x, diperoleh (1-n) EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3 y EMBED Equation.3 =  EMBED Equation.3  EMBED Equation.3  Substitusikan y EMBED Equation.3  = v dan y EMBED Equation.3 =  EMBED Equation.3  EMBED Equation.3  ke persamaan y EMBED Equation.3  + P(x) y EMBED Equation.3  = Q(x) diperoleh  EMBED Equation.3  EMBED Equation.3  + P(x)v = Q(x)  EMBED Equation.3  EMBED Equation.3  + (1-n)P(x) v = (1-n)Q(x) Bentuk terakhir adalah persamaan differensial linear yang selesaian umum dapat dicari dengan metode faktor integral atau metode Lagrange atau metode Pengubahan persamaan differensial eksak. Misal (1-n)P(x) = p(x) dan (1-n)Q(x) = q(x) Maka selesaian umumnya adalah v = e EMBED Equation.3  ( EMBED Equation.3  q(x) e EMBED Equation.3  dx ) Contoh soal Tentukan selesaian umum persamaan 1.  EMBED Equation.3 - y = xy EMBED Equation.3  Jawab  EMBED Equation.3 - y = xy EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  - y EMBED Equation.3  = x Misal y EMBED Equation.3  = v maka -2y EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  = -  EMBED Equation.3  Substitusikan ke persamaan semula, didapat: - EMBED Equation.3  - v = x  EMBED Equation.3   EMBED Equation.3  + 2v = -2x dimana p(x) = 2 , q(x) = -2x dan faktor integral (I) = e EMBED Equation.3 = e EMBED Equation.3  EMBED Equation.3  selesaian umumnya ve EMBED Equation.3  =  EMBED Equation.3 q(x)e EMBED Equation.3  dx ve EMBED Equation.3 =  EMBED Equation.3 dx  EMBED Equation.3  EMBED Equation.3 = -xe EMBED Equation.3 + e EMBED Equation.3 + c 2.  EMBED Equation.3 + y = y EMBED Equation.3 (Cos x - Sin x) Jawab  EMBED Equation.3  EMBED Equation.3  + y EMBED Equation.3  = (cos x - sin x)  EMBED Equation.3  EMBED Equation.3  + y EMBED Equation.3  = (cos x - sin x) Misal y EMBED Equation.3  = v maka -y EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  = -  EMBED Equation.3  Substitusikan ke persamaan semula, didapat: - EMBED Equation.3  + v = (cos x - sin x)  EMBED Equation.3   EMBED Equation.3  - v = -(cos x - sin x) dimana p(x) = -1 , q(x) = (sin x - cos x) dan factor integral e EMBED Equation.3 = e EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  selesaian umumnya ve EMBED Equation.3  =  EMBED Equation.3 q(x)e EMBED Equation.3  dx ve EMBED Equation.3 =  EMBED Equation.3 dx  EMBED Equation.3  EMBED Equation.3 = -e EMBED Equation.3 sin x + C adalah selesaian umumnya. Soal-soal Selidiki apakah persamaan differensial tingkat satu derajat satu dibawah ini termasuk persamaan differensial linear. 1.  EMBED Equation.3 - 2y = 3 3x 2. x dy 2y dx = (1+x)e EMBED Equation.3 dx 3. y dx + (xy + x 3y) dy = 0,  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3 , P(y) = 1, Q(y) = 3 -  EMBED Equation.3 , y dx + (xy + x 3y) dy = 0,  EMBED Equation.3  4. x dy y dx = x EMBED Equation.3  dy 5. (1+sin y) dx = [2y cos y x(sec y + tgn y)] dy 6. (2xy EMBED Equation.3 y) dx + 2x dy = 0 7.  EMBED Equation.3 + 2xy = 5x EMBED Equation.3 y EMBED Equation.3  8. (x EMBED Equation.3 +1)  EMBED Equation.3  + xy = x 9.  EMBED Equation.3  - y = y EMBED Equation.3 x 10. xy = 2y + x EMBED Equation.3 e EMBED Equation.3  EMBED Equation.3  Tentukan selesaian umum persamaan linear berikut dengan menggunakan cara yang sesuai. 1.  EMBED Equation.3 + 3xy = 2 2.  EMBED Equation.3  =  EMBED Equation.3  + 2x + 1 Jawab  EMBED Equation.3  =  EMBED Equation.3  + 2x + 1  EMBED Equation.3  Didapat P(x) =  EMBED Equation.3 , Q(x) = (2x+1) Faktor integral (I) = e EMBED Equation.3  = e EMBED Equation.3  = e EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  Primitif dari  EMBED Equation.3  =  EMBED Equation.3  + 2x + 1 adalah Iy =  EMBED Equation.3   EMBED Equation.3 y = x EMBED Equation.3   EMBED Equation.3 y = x  EMBED Equation.3   EMBED Equation.3 y = x(2x +ln EMBED Equation.3   EMBED Equation.3  y = 4x + (ln EMBED Equation.3 +1) +C 3.  EMBED Equation.3 + 2xy = 5y EMBED Equation.3  4.  EMBED Equation.3 + 3y = 3x EMBED Equation.3 e  EMBED Equation.3  5. Cos  EMBED Equation.3  dr + ( r sin  EMBED Equation.3 - cos EMBED Equation.3  EMBED Equation.3 ) d EMBED Equation.3 = 0 6. y EMBED Equation.3 dx + (3xy-1) dy = 0  EMBED Equation.3  7. r dt 2t dr = (r-2)e EMBED Equation.3 dr 8.  EMBED Equation.3 - 6y = 10 sin 2x P(x) = -6, Q(x) = 10 sin 2x I = e EMBED Equation.3  Primitifnya Iy =  EMBED Equation.3 dx y = e EMBED Equation.3  = 10 e EMBED Equation.3  e EMBED Equation.3  = -5 cos 2x 15 sin 2x 15e EMBED Equation.3  EMBED Equation.3  25 EMBED Equation.3  = -5 cos 2x 15 sin 2x  EMBED Equation.3  = EMBED Equation.3  (-5 cos 2x 15 sin 2x+C) Didapatkan primitif Y =  EMBED Equation.3  (-5 cos 2x 15 sin 2x+C) 9. dy + (2y cos x + sin 2x) dx = 0 10. (1+y EMBED Equation.3 ) dx = ( arc tan y x) dy Tentukan selesaian masalah nilai awal 1.  EMBED Equation.3 -  EMBED Equation.3 = xe EMBED Equation.3  dengan y(1) =  EMBED Equation.3  2.  EMBED Equation.3  4y - e EMBED Equation.3  = 0 dengan y(0) =  EMBED Equation.3  3.  EMBED Equation.3 xy = sin 2t dengan x(0) = 0 4.  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