ABSTRACT
So far we have studied ODEs where the dependent variable is a scalar function. In particular, we have discussed various methods of finding their solutions. Moreover, we have studied the quantitative and qualitative behavior of the solutions of these equations in case we are unable to find them explicitly. In this chapter, we discuss some methods to solve a system of linear ODEs. Let https://www.w3.org/1998/Math/MathML"> J ⊆ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003153757/8dd02ec3-2a7d-4f0c-8e17-221309973715/content/math05_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be an open interval, https://www.w3.org/1998/Math/MathML"> Ω ⊆ ℝ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003153757/8dd02ec3-2a7d-4f0c-8e17-221309973715/content/math05_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> an open set. We assume that https://www.w3.org/1998/Math/MathML"> F : J ¯ × Ω ¯ → ℝ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003153757/8dd02ec3-2a7d-4f0c-8e17-221309973715/content/math05_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a continuous function, https://www.w3.org/1998/Math/MathML"> Y ( t ) = ( y 1 ( t ) , … , y n ( t ) ) , t ∈ J . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003153757/8dd02ec3-2a7d-4f0c-8e17-221309973715/content/math05_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Our objective is to study the following system https://www.w3.org/1998/Math/MathML"> Y ′ ( t ) = F ( t , Y ( t ) ) , t ∈ J , Y ( t 0 ) = Y 0 ∈ ℝ n . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003153757/8dd02ec3-2a7d-4f0c-8e17-221309973715/content/math05_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
