In a radioactive series consisting of four different nuclides starting with the parent substance N₁ and ending with the stable product N, the amounts of each nuclide present at time t are given by the differential equations model dN₁ dr = -6N₁ dN₂ dt = 6N₁ - 4N₂ dN3 dt = 4N₂ - 2N3 dN₂ 2N3 dr Express these in the vector-matrix form N = AN where N=[N₁ N₁ N₁ N₁]. Find the eigenvalues and corresponding eigenvectors of A. Using the spectral form of the solution, determine N.(t) given that at time t = 0, N₁ = C and N₂ = N₁-N₂ = 0.
In a radioactive series consisting of four different nuclides starting with the parent substance N₁ and ending with the stable product N, the amounts of each nuclide present at time t are given by the differential equations model dN₁ dr = -6N₁ dN₂ dt = 6N₁ - 4N₂ dN3 dt = 4N₂ - 2N3 dN₂ 2N3 dr Express these in the vector-matrix form N = AN where N=[N₁ N₁ N₁ N₁]. Find the eigenvalues and corresponding eigenvectors of A. Using the spectral form of the solution, determine N.(t) given that at time t = 0, N₁ = C and N₂ = N₁-N₂ = 0.
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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![In a radioactive series consisting of four different
nuclides starting with the parent substance N₁ and
ending with the stable product N, the amounts of
each nuclide present at time t are given by the
differential equations model
dN₁
dt
=-6N₁
dN₂ =
= 6N₁ - 4N₂
dt
dN
= 4N₂ - 2N₂
dt
dN₁
= 2N₂
dt
Express these in the vector-matrix form
N = AN
where N = [N₁ N₂ N3 N4]. Find the eigenvalues
and corresponding eigenvectors of A. Using the
spectral form of the solution, determine N₂(1) given
that at time t = 0, N₁ = C and N₂ = N₂= N₁ = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8edaa379-612e-4a19-9b47-c754f48a7789%2F42c01a09-ecf6-4bd2-be60-884aac7d201c%2F5nqs5xk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In a radioactive series consisting of four different
nuclides starting with the parent substance N₁ and
ending with the stable product N, the amounts of
each nuclide present at time t are given by the
differential equations model
dN₁
dt
=-6N₁
dN₂ =
= 6N₁ - 4N₂
dt
dN
= 4N₂ - 2N₂
dt
dN₁
= 2N₂
dt
Express these in the vector-matrix form
N = AN
where N = [N₁ N₂ N3 N4]. Find the eigenvalues
and corresponding eigenvectors of A. Using the
spectral form of the solution, determine N₂(1) given
that at time t = 0, N₁ = C and N₂ = N₂= N₁ = 0.
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