assignment 8 math 2280
TRANSCRIPT
Math 2280 - Assignment 8
Dylan Zwick
Spring 2013
Section 5.4 - 1, 8, 15, 25, 33
Section 5.5 - 1, 7, 9, 18, 24
Section 5.6 - 1, 6, 10, 17, 19
1
Section 5.4 - Multiple Eigenvalue Solutions
5.4.1 - Find a general solution to the system of differential equations be‘ow.
The
I.-
(_(
Aje
I 1 / 1 J_ tooLH } - 0 0
/ ‘ICc) WoYK
Ix’=(l 4)x
a5 )7evq /tj:
(>3)2
I7eflv/ej-
C+Cj JcI10 )
il e ci’ e
(hct(Y
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(4e yY/ O/f
\1)( )2
5.4.8 Find a general solution to the system of differential equations below.
/25 12 Ox’=f —18 —5 0 )x
\\ 6 6 13)
Z5 j o-y 0
6 lA
F$, /e; 7. 15 II
f / —7 Q elyekl ec7I
: ):)(-it)
0/
) 3 ‘‘ Ice e (7 eji ec/c1 5
IL Q
Kb) : 0
y e / 5c’ 10 I C)7 I
()7()e1frI3f
5.4.25 - Find a general solution to the system of differential equations below. The eigenvalues of the matrix are given.
/ —2 17 4 \x’= f —1 6 1 ) x; A = 2.2.2.
0 1 2)
€ie cri
(7 L)
1 -1 b =(
I7QV(ec4cr4 ( )3 c) /ic(
2iVCd
Lj 1( H /1 H U to 00\ 0 C) -) c
7
5.4.33 - The characteristic equation of the coefficient matrix A of the system
3 —4 1 043010 0 3 40043
is
= (A2 — 6.X + 25)2 = 0.
Therefore, A has the repeated complex conjugate pair 3+ 4i of eigenvalues. First show that the complex vectors
1 0i 010 10 i
form a length 2 chain {vi. v2} associated with the eigenvalue A =
3 — 4i. Then calculate the real and imaginary parts of the complex-valued solutions
v1e and (vitfv9)eAt
to find four independent real-valued solutions of x’ = Ax.
‘Note in the textbook there’s a typo in this vector.
9
Matrix Exponentials and Linear Systems
5.5.1 - Find a fundamental matrix for the system below, and then find asolution satisfying the given initial conditions.
2 ix’1 9)x. x(O)=(32).
1ve £ciI e ei7eyivci/j of fle
+j (A-i)(A-))I zf
\:
k1 eIj1vco( #cI iNI
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P 7 a y e fl V c ic / t7
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Nonhomogeneous Linear Systems
5.6.1 - Apply the method of undetermined coefficients to find a particularsolution to the system below.
= x + 2y + 3= 2x + y — 2
41 1
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