faseportrait-ziyadatur (12610007)
5
Nama: Ziyadatur R.F. NIM: 12610007 Misalkan matriks A = [ a b p q ] = [ 7 1 6 2 ] dx dt =ax +by dx dt =7 x+ y dy dt =px +qy dy dt =6 x+ 2 y Mencari nilai Eigen [ x' y' ] = [ 7 1 6 2 ][ x y ] det ( A −λI) =0 det ( [ 7 1 6 2 ] − λ [ 1 0 0 1 ] ) =0 det ( [ 7 1 6 2 ] − [ λ 0 0 λ ] ) =0 det ( [ 7−λ 1 6 2−λ ] ) =0 ( 7− λ )( 2−λ ) −6=0 14 −2 λ−7 λ +λ 2 =0 λ 2 − 9 λ +8=0
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Transcript of faseportrait-ziyadatur (12610007)
Nama: Ziyadatur R.F.
NIM: 12610007
Misalkan matriks
A=[ a bp q]=[7 1
6 2] dxdt
=ax+by
dxdt
=7 x+ y
dydt
=px+qy
dydt
=6 x+2 y
Mencari nilai Eigen
[ x 'y ' ]=[7 16 2][ xy ]
det (A−λI )=0
det ([7 16 2]−λ [1 0
0 1 ])=0
det ([7 16 2]−[λ 0
0 λ ])=0
det ([7−λ 16 2−λ])=0
(7−λ ) (2− λ )−6=0
14−2 λ−7 λ+λ2=0
λ2−9 λ+8=0
( λ−1 )(λ−8)
λ1=1 dan λ2=8
Untuk λ1=1
Au=λu
[7 16 2][u1
u2]=1 [u1
u2]
7u1+u2=u1
6u1+2u2=u2
6u1=−u2
6u1=−u2
u1
u2
= 1−6
Maka [u1
u2]=[ 1
−6]
Untuk λ2=8
Av=λ v
[7 16 2][v1
v2]=1[v1
v2]
7 v1+v2=8v1
6 v1+2 v2=8v2
−v1=−v2
6 v1=6v2
v1
v2
=66
Maka [v1
v2]=[11]
Misalkan
x (0 )=7 dan y (0 )=7
( x 'y ')=C1(u1
u2)eλt−C2(v1
v2)e λt
( x 'y ')=C1( 1−6)e t−C2(11)e8 t
Maka
(77)=C1( 1−6)e(0 )−C2(1
1)e(0)
7=C1−C2
7=−6C1−C2
Maka didapatkan C1=−14
5 dan C2=
−495
Sehingga didapatkan solusi umumnya adalah
( xy )=−145 ( 1
−6)e t+ 495 (11)e8 t
Program matlab
restart;
with(DEtools);
phaseportrait([diff(X(t), t) = 7*X(t)-Y(t), diff(Y(t), t) = 6*X(t)+2*Y(t)], [X(t), Y(t)], t = 0 .. 10, {[X(0) = 7, Y(0) = 7]});
phaseportrait([diff(X(t), t) = 7*X(t)-Y(t), diff(Y(t), t) = 6*X(t)+2*Y(t)], [X(t), Y(t)], t = 0 .. 10, {[X(0) = -7, Y(0) = -7]});
phaseportrait([diff(X(t), t) = 7*X(t)-Y(t), diff(Y(t), t) = 6*X(t)+2*Y(t)], [X(t), Y(t)], t = 0 .. 10, {[X(0) = -7, Y(0) = 7]});
phaseportrait([diff(X(t), t) = 7*X(t)-Y(t), diff(Y(t), t) = 6*X(t)+2*Y(t)], [X(t), Y(t)], t = 0 .. 10, {[X(0) = 7, Y(0) = -7]});
