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1. Use Theorem 1 to obtain the general solution of the given first-order systems; if initial conditions are given, find the particular solution satisfying them. (a) x' = (c) x' = 5 2 4 2 X; 0 -1 -2 -4 -2 x; (b) x' = (d) x' 1 X, 9 -6 6 x(0) = []; 4 -1 ; Sol'n 0 x, x(0) 4 3 -4. =

1. Use Theorem 1 to obtain the general solution of the given first-order systems; if initial conditions are given, find the particular solution satisfying them. (a) x' = (c) x' = 5 2 4 2 X; 0 -1 -2 -4 -2 x; (b) x' = (d) x' 1 X, 9 -6 6 x(0) = []; 4 -1 ; Sol'n 0 x, x(0) 4 3 -4. =

Advanced Engineering Mathematics
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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Please show all steps clearly, thank you. I only need help with problem 1.d.

Exercises 1. Use Theorem 1 to obtain the general solution of the given first-order systems; if initial conditions are given, find the particular solution satisfying them. [23] (a) x' (c) x' = 4 0 X; 1 -2 (b) x² = 6 -2 x; (d) x' -4 1 4 x, x(0): = 4 0 9 -6 -1 6 4 8₁ ; Sol'n 0 x, x(0) 3 = 1 H
Transcribed Image Text:Exercises 1. Use Theorem 1 to obtain the general solution of the given first-order systems; if initial conditions are given, find the particular solution satisfying them. [23] (a) x' (c) x' = 4 0 X; 1 -2 (b) x² = 6 -2 x; (d) x' -4 1 4 x, x(0): = 4 0 9 -6 -1 6 4 8₁ ; Sol'n 0 x, x(0) 3 = 1 H
Theorem 1. If A is an (n×n)-matrix of real constants that has an eigenbasis V₁, . . . , Vn for R", then the general solution of (7.17) is given by x(t) = C₁ e¹¹tv₁ + ... + C₂ e Ant√n₂ where A₁,..., An are the (not necessarily distinct) eigenvalues associated respectively with the eigenvectors V₁,..., Vn.
Transcribed Image Text:Theorem 1. If A is an (n×n)-matrix of real constants that has an eigenbasis V₁, . . . , Vn for R", then the general solution of (7.17) is given by x(t) = C₁ e¹¹tv₁ + ... + C₂ e Ant√n₂ where A₁,..., An are the (not necessarily distinct) eigenvalues associated respectively with the eigenvectors V₁,..., Vn.
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