Attached is a proof of the general solution to a system of differential equations that has secular terms as a result of repeated eigenvalues, and hence solved using a Jordan Normal form.
I can follow the proof fine, however the proof claims to be, and is clearly 'inductive' in nature, but i'm struggling to formalise it as a standard "proof by induction". By standard I mean where we state $P(1)$ is true, let's assume $P(n)$ is true and hence show $P(n+1)$ is true. Or $n - 1 \implies n$ is true.
I have attached a screenshot as I don't want to paraphrase and retyping would be unnecessarily pain staking...I hope this is okay and please see the screenshot!
I have corrected a small subscript typo in the text, and highlighted what are the key pieces of information that i'm trying to work into the formal induction structure. My current two attempts are below. Thank you!
Theory 1
We are proving for $n$ cases and we take $k-j$ as our $n^{th}$ term .
This allows us to say that $n-1$ cases corresponds to $k-j-1 = k-(j+1)$
This fits with some of their notation and allows us to assume that $j+1$ is true as they do.
Where our usual $P(1)$ here i guess would be with $j = 0$ hence giving us the $w'_k = λw_k \implies w_k = c_ke^{λt}$ they have written. This is obviously not right because $j \in [1,(k-1)]$ but it's an idea.
Theory 2
Our $n$ cases is $k$ and so then we assume $k-1$ is true, we actually use strong induction stating that it's true for all $j \in [1,(k-1)]$ and this allows us to use our $j+1$
So we assume $P(k-1)$ and go on to prove $P(k)$
Our $P(1)$ i suppose would just be $w'_1 = λw_1 \implies w_1 = c_1e^{λt}$
But they haven't written this.
Help greatly appreciated, thanks!
