System of linear differential equations

Given System of linear differential equations with constant coefficient:


The matrix coefficient :


And also given that the vector :


Is a solution to the system.
1.Show that x1(t),x2(t),...,xn(t) are differetiable from every order in interval I and also that:
\[\frac{d^k \mathbf{x}(t)}{dt^k} = \mathbf{A}^k \mathbf{x}(t),\]
2.Show that there Is System of linear differential equations with constant coefficient from order n such that 
x1(t),..,xn(t) are a solution. Hint: Kylie Hamilton theorem 

Ordinary Differential Equations
Danielatia123 Danielatia123
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Martin Martin
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  • Danielatia123 Danielatia123
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    I understand why we can differentiate one time, but what is the justification for second, third, and go on?

    • Martin Martin
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      It's by mathematical induction. The step from n-1 to n allows to go from 2 to 3 to 4 to ... To be rigorous you only need the step from 1 to 2 (the base case) and from n-1 to n (the induction step).

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