<s>[INST] system of differential linear equations $y'=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}y$ find the solution to the problem $y'=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}y, y(0)=\begin{pmatrix}4\\0\end{pmatrix}$

I know i have to find the eigenvalues and eigenvectors of the matrix  $A=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$
there's only one eigenvalue which is $1$ and only "one" eigenvector and we can choose $\begin{pmatrix} 1\\0\end{pmatrix}$
but now I dont know what to do.
what comes next?
I should say that I know how to find the exponencial of a matrix and also I want to know how to solve this kind of problem in general, so techniques to solve this particular one that wouldnt work on a more general problem dont help me much.
 [/INST] Let
$$I_2 = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\quad\text{ and }\quad J_2 = \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix}$$
Notice $I_2 J_2 = J_2 I_2$, $I_2^2 = I_2$ and $J_2^2 = 0_2$, we have:
$$e^{At} = e^{I_2t + J_2t} = e^{I_2t}e^{J_2t} = (I_2 + I_2t + I_2^2\frac{t^2}{2!} + \cdots)( I_2 + J_2t+ J_2^2\frac{t^2}{2}+\cdots)\\= I_2( 1 + t + \frac{t^2}{2} + \cdots)(I_2+J_2t) = e^t (I_2+J_2t) = e^t\begin{pmatrix}1 & t\\0 & 1\end{pmatrix}$$
This gives us:
$$y(t) = e^{At}y(0) = e^t \begin{pmatrix}1 & t\\0 & 1\end{pmatrix}\begin{pmatrix}4\\0\end{pmatrix} = \begin{pmatrix}4 e^t\\0\end{pmatrix}$$
The same approach allow you to deal with other linear ODE: $y' = A y$ where $A$ is not diagonalizable. Let's say you have use a similarity transform and bring $A$ into a Jordan
normal block. For each Jordan block of size $n$:
$$B = \begin{pmatrix}
\lambda & 1 & 0 &\ldots & 0\\
0 & \lambda & 1 &\ldots & 0\\
  &   &  \ddots & \ddots & &\\
0 & 0 & \ldots & \lambda & 1\\
0 & 0 & 0 &\ldots & \lambda
\end{pmatrix}$$
You can rewrite $B$ as $\lambda I_n + J_n$ where $J_n$ is a $n\times n$ matrix with entries on the superdiagonal all equal to 1. Once again, $I_n J_n = J_n I_n$ and $J_n^n = 0_n$ and one has in general:
$$e^{Bt} = e^{(\lambda I_n + J_n)t} = e^{\lambda t} \left(1 + J_n t + J_n^2\frac{t^2}{2!} + \cdots J_n^{n-1}\frac{t^{n-1}}{(n-1)!}\right)
= e^{\lambda t} 
\begin{pmatrix}
1 & t & \frac{t^2}{2!} &\ldots & \frac{t^{n-1}}{(n-1)!}\\
0 & 1 & t &\ldots & \frac{t^{n-2}}{(n-2)!}\\
  &   &  & \ddots & &\\
0 & 0 & \ldots & 1 & t\\
0 & 0 & 0 &\ldots & 1
\end{pmatrix}$$
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