Find the general solution of the given system of equations and then describe how the solutions behave as t→∞. x' x = (¹5-20) x(t) x(t) = x(t) x(t) = = Hint = x(t) = Clest C₁[test стел C₁ est (²) + C₁ est (²) (²) + C₂ [test eTextbook and Media (²1) + e²¹ (1) ₁² est ; solution approaches the critical point x = 0. + c₂test (61) + ³¹ (²) 1² est + c₂[test (²); (²³) + ³¹ (1) est X ]; solution surrounds the critical point x = 0. ]; solution becomes unbounded as t→∞. ; solution becomes unbounded as t→∞. ]; solution becomes unbounded as t→∞. If A has a single repeated eigenvalue μ, then a general solution of x' = Ax is x=c₁e¹v₁+c₂e¹t√₂ if v₁ and v₂ are independent eigenvectors, or C₁e¹tv + c₂e¹t (w+tv) where (A-AI)w = v if v is the only eigenvector of A.
Find the general solution of the given system of equations and then describe how the solutions behave as t→∞. x' x = (¹5-20) x(t) x(t) = x(t) x(t) = = Hint = x(t) = Clest C₁[test стел C₁ est (²) + C₁ est (²) (²) + C₂ [test eTextbook and Media (²1) + e²¹ (1) ₁² est ; solution approaches the critical point x = 0. + c₂test (61) + ³¹ (²) 1² est + c₂[test (²); (²³) + ³¹ (1) est X ]; solution surrounds the critical point x = 0. ]; solution becomes unbounded as t→∞. ; solution becomes unbounded as t→∞. ]; solution becomes unbounded as t→∞. If A has a single repeated eigenvalue μ, then a general solution of x' = Ax is x=c₁e¹v₁+c₂e¹t√₂ if v₁ and v₂ are independent eigenvectors, or C₁e¹tv + c₂e¹t (w+tv) where (A-AI)w = v if v is the only eigenvector of A.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Find the general solution of the given system of equations and then describe how the solutions behave as t→∞.
20
- (15 -29)x
X
5
x(t) =
x(t)
x(t)
=
+ C₂ [test
(²)
(²3) + ~*² (6) ₁
(6)
est
x(t) = C₁ est
=
Hint
C₁ est
C₁[test
C₁ eSt
x(t) = c₁e5t
(²);
(²)
(²)
eTextbook and Media
+ c₂test
+
+ est
; solution approaches the critical point x = 0.
(²)
(²)
C₂ [test
(²)₁
X' =
]; solution becomes unbounded as t→∞.
]; solution surrounds the critical point x = 0.
+est
; solution becomes unbounded as t→∞.
(6) ¹:
]; solution becomes unbounded as t→∞.
If A has a single repeated eigenvalue μ, then a general solution of x' = Ax is
x=c₁e¹t√₁+c₂e¹t√₂ if v₁ and v₂ are independent eigenvectors, or
C₁e¹tv + c₂e¹¹ (w+tv) where (A-AI)w=v_if v is the only eigenvector of A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89b1189c-c32b-4144-b592-a832138b3e54%2F24d79d6d-3dde-4bd2-8814-13ef17bb3c2a%2Fui694f1_processed.png&w=3840&q=75)
Transcribed Image Text:Find the general solution of the given system of equations and then describe how the solutions behave as t→∞.
20
- (15 -29)x
X
5
x(t) =
x(t)
x(t)
=
+ C₂ [test
(²)
(²3) + ~*² (6) ₁
(6)
est
x(t) = C₁ est
=
Hint
C₁ est
C₁[test
C₁ eSt
x(t) = c₁e5t
(²);
(²)
(²)
eTextbook and Media
+ c₂test
+
+ est
; solution approaches the critical point x = 0.
(²)
(²)
C₂ [test
(²)₁
X' =
]; solution becomes unbounded as t→∞.
]; solution surrounds the critical point x = 0.
+est
; solution becomes unbounded as t→∞.
(6) ¹:
]; solution becomes unbounded as t→∞.
If A has a single repeated eigenvalue μ, then a general solution of x' = Ax is
x=c₁e¹t√₁+c₂e¹t√₂ if v₁ and v₂ are independent eigenvectors, or
C₁e¹tv + c₂e¹¹ (w+tv) where (A-AI)w=v_if v is the only eigenvector of A.
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