2. Given a system of first order linear differential equations [] = [2] *₁ = [ ²5 ] then this system (2) can also be solved by manipulating the properties of matrix exponential. In this group problem-solving process, you will be guided to employ this approach: -4 a) By letting A = 3], first, each team member should be (fully) convinced that the eigenvalues of A are A1,2 = 5 (and let us call it A). Next, by writing C = AI and B = A-AI, describe some steps on how you can show that BC = CB. b) Now, suppose that e At = eBt. ect, work out the matrix e Ct in terms of I. 2 (Hints: Use e* = 1+x++ A³ + ... Then, try to deduce the patterns for A², A³ and etc). 3! ... and generalise this to a matrix A by defining e4 = 1 + A +₁₁+ c) Employ the same techniques as in part (b) i.e., by scrutinising the patterns for B2, B³ etc and using the definition for ex, propose a single matrix for eBt. d) Notice that the solution is given by x(t) = eAtxo = eBt. eCtxo. Using this fact, work out the solution of system (2), x (t).
2. Given a system of first order linear differential equations [] = [2] *₁ = [ ²5 ] then this system (2) can also be solved by manipulating the properties of matrix exponential. In this group problem-solving process, you will be guided to employ this approach: -4 a) By letting A = 3], first, each team member should be (fully) convinced that the eigenvalues of A are A1,2 = 5 (and let us call it A). Next, by writing C = AI and B = A-AI, describe some steps on how you can show that BC = CB. b) Now, suppose that e At = eBt. ect, work out the matrix e Ct in terms of I. 2 (Hints: Use e* = 1+x++ A³ + ... Then, try to deduce the patterns for A², A³ and etc). 3! ... and generalise this to a matrix A by defining e4 = 1 + A +₁₁+ c) Employ the same techniques as in part (b) i.e., by scrutinising the patterns for B2, B³ etc and using the definition for ex, propose a single matrix for eBt. d) Notice that the solution is given by x(t) = eAtxo = eBt. eCtxo. Using this fact, work out the solution of system (2), x (t).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![2. Given a system of first order linear differential equations
[] = [2] = [²],
*₁
then this system (2) can also be solved by manipulating the properties of matrix exponential. In this
group problem-solving process, you will be guided to employ this approach:
a) By letting A = 3], first, each team member should be (fully) convinced that the eigenvalues
-4
of A are A1,2 = 5 (and let us call it A). Next, by writing C = AI and B = A-AI, describe some steps on how
you can show that BC = CB.
b) Now, suppose that eAt = eBt. ect, work out the matrix e Ct in terms of I.
x2
(Hints: Use e* = 1+x++
2!
...
and generalise this to a matrix A by defining e4 = 1 + A +₁₁+
A³
+ ... Then, try to deduce the patterns for A², A³ and etc).
3!
c) Employ the same techniques as in part (b) i.e., by scrutinising the patterns for B2, B3 etc and using
the definition for ex, propose a single matrix for eBt.
d) Notice that the solution is given by x(t) = eAtxo = eBt. eCtxo. Using this fact, work out the
solution of system (2), x (t).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60abd4de-ba2d-4a53-b8fc-e1d2e0799cec%2F9e198784-834d-45af-979e-742df3f31eae%2F8pqb2aw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Given a system of first order linear differential equations
[] = [2] = [²],
*₁
then this system (2) can also be solved by manipulating the properties of matrix exponential. In this
group problem-solving process, you will be guided to employ this approach:
a) By letting A = 3], first, each team member should be (fully) convinced that the eigenvalues
-4
of A are A1,2 = 5 (and let us call it A). Next, by writing C = AI and B = A-AI, describe some steps on how
you can show that BC = CB.
b) Now, suppose that eAt = eBt. ect, work out the matrix e Ct in terms of I.
x2
(Hints: Use e* = 1+x++
2!
...
and generalise this to a matrix A by defining e4 = 1 + A +₁₁+
A³
+ ... Then, try to deduce the patterns for A², A³ and etc).
3!
c) Employ the same techniques as in part (b) i.e., by scrutinising the patterns for B2, B3 etc and using
the definition for ex, propose a single matrix for eBt.
d) Notice that the solution is given by x(t) = eAtxo = eBt. eCtxo. Using this fact, work out the
solution of system (2), x (t).
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