2. Exercise §3.4 #7. Let A be a positive-definite n z matrix. Let S(t)=(-1) 42m2m+1 (2m + 1)! m=0 (a) Show that this series of matrices converges uniformly for bounded t and its sum S(t) solves the problem S"(t) + A²S(t) = 0, S(0) = 0, S'(0) = I, where I is the identity matrix. Therefore, it makes sense to denote S(t) as A-¹ sint A and to denote its derivative S'(t) as cos(tA).

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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2. Exercise §3.4 #7. Let A be a positive-definite n z matrix. Let
8 (-1)m 42m+2m+1
(2m + 1)!
S(t)=
m=0
(a) Show that this series of matrices converges uniformly for bounded t and its sum S(t) solves the
problem S" (t) + A²S(t) = 0, S(0) = 0, S'(0) = I, where I is the identity matrix. Therefore, it makes
sense to denote S(t) as A-¹ sint A and to denote its derivative S'(t) as cos(tA).
Transcribed Image Text:2. Exercise §3.4 #7. Let A be a positive-definite n z matrix. Let 8 (-1)m 42m+2m+1 (2m + 1)! S(t)= m=0 (a) Show that this series of matrices converges uniformly for bounded t and its sum S(t) solves the problem S" (t) + A²S(t) = 0, S(0) = 0, S'(0) = I, where I is the identity matrix. Therefore, it makes sense to denote S(t) as A-¹ sint A and to denote its derivative S'(t) as cos(tA).
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