Reference book for linear differential equation in the light of linear algebra.

Question

Usually in undergraduate course student is introduced to mechanical things of differential equation consisting of calculations,integrations,and different methods or tricks of solving the equations without mentioning why we do so.For example in case of linear differential equation with constant coefficients,we first solve the reduced equation to find a complementary function and then find a particular integral for the equation.We are only told that in this way you would get the solution but not told why we are doing it e.g. that $F(D)$ is a linear operator and we are first finding kernel space of the operator by computing the complementary function.These things include a basis knowledge of linear algebra which I have now.So I want to understand properly the linear algebra aspects behind the methods of differential equation solving especially linear ones.Can anyone suggest me some reference book where I can find the reasons behind what we are doing to solve a linear differential equation(in light of linear algebra).I want to look at differential equations analytically not mechanically.I have already seen Hirsch Smale book which discusses on it but it does not fully satisfy my purpose.

Answer 1

Hirch Smale' book is good.One can try also Arnold's textbook.