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2. (a) Prove that if x1,...,x, are a fundamental set of solutions of the linear homoge- neous system x' = A(t)x then for any c1,..., Cn E R, %3D X = C1X1 + + CnXn is also a solution. (It is not required here that the solutions form an FSS.) (b) Now prove that if y is a solution of the non-homogeneous system x' = A(t)x+g(t) then x.+ y is also a solution. %3D (c) Finally prove that any solution of the non-homogeneous system must be of the form x. +y for some c1, , Cn

2. (a) Prove that if x1,...,x, are a fundamental set of solutions of the linear homoge- neous system x' = A(t)x then for any c1,..., Cn E R, %3D X = C1X1 + + CnXn is also a solution. (It is not required here that the solutions form an FSS.) (b) Now prove that if y is a solution of the non-homogeneous system x' = A(t)x+g(t) then x.+ y is also a solution. %3D (c) Finally prove that any solution of the non-homogeneous system must be of the form x. +y for some c1, , Cn

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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2. (a) Prove that if x1,...,xn are a fundamental set of solutions of the linear homoge- neous system x' A(t)x then for any c1,..., Cn E R, %3D Xe = C1X1 +...+ CnXn is also a solution. (It is not required here that the solutions form an FSS.) (b) Now prove that if y is a solution of the non-homogeneous system x' = A(t)x+g(t) then xe+y is also a solution. (c) Finally prove that any solution of the non-homogeneous system must be of the form x. +y for some c1, ., Cn.
Transcribed Image Text:2. (a) Prove that if x1,...,xn are a fundamental set of solutions of the linear homoge- neous system x' A(t)x then for any c1,..., Cn E R, %3D Xe = C1X1 +...+ CnXn is also a solution. (It is not required here that the solutions form an FSS.) (b) Now prove that if y is a solution of the non-homogeneous system x' = A(t)x+g(t) then xe+y is also a solution. (c) Finally prove that any solution of the non-homogeneous system must be of the form x. +y for some c1, ., Cn.
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