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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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