(i) Combine Exercises 10.1.2324 to show that if A = SS 1 is diagonalizable, then the solution...
Question:
(i) Combine Exercises 10.1.23–24 to show that if A = SS−1 is diagonalizable, then the solution to u̇ = Au can be written as u(t) = S (c1 eλ1 t, . . . , cn eλn t)T, where λ1, . . . , λn are its eigenvalues and S = (v1 v2 . . . vn) is the corresponding matrix of eigenvectors.
Data From 10.1.23
Prove that the general solution to a linear system u̇ = Λu with diagonal coefficient a matrix Λ = diag (λ1, . . . , λn) is given by u(t) = (c1 eλ1 t, . . . , cn eλnt)T.
Data From 10.1.24
Show that if u(t) is a solution to u̇ = Au, and S is a constant, nonsingular matrix of the same size as A, then v(t) = Su(t) solves the linear system v̇ = Bv, where B = SAS−1 is similar to A.(ii) Write the general solution to the systems in Exercise 10.1.13 in this form.
Data From Exercise 10.1.13
Solve the following initial value problems:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Step by Step Answer: