What was the basis for the repeated-eigenvalue formula? Long ago, I was curious and I was wondering if any professionals here had any input. I was curious how it was conjectured that the general solution of a linear system with a repeated eigenvalue would take the following form:
$Y(t) = (e^λ)^t(x_o,y_o)+t(e^λ)^t(x_o*y_o, x_o*y_o)$, where $*$ is an operation between X and Y. 
It's a very delicate system and I've tried reading through various DE texts and history of mathematics texts..but could not find an answer.
 A: You can also see this with integrating factors. For example, if $D$ is the differential operator $\frac{d}{dt}$, then
$$
          (D-\lambda)g = e^{\lambda t}De^{-\lambda t}g.
$$
Therefore, for any positive integer $n$,
$$
                 (D-\lambda)^{n}g = e^{\lambda t}D^{n}e^{-\lambda t}g
$$
That means a solution of $(D-\lambda)^{n}g=0$ must satisfy
$$
                 \frac{d^{n}}{dt^{n}}(e^{-\lambda t}g)=0,
$$
the general solution of which is a polynomial
$$
                      (e^{-\lambda t}g) = a_{0}+a_{1}t+\cdots a_{n-1}t^{n-1}.
$$
Any linear differential equation with constant coefficients can be solved as sums of such terms because of how the factors in $(D-\mu)$ commute. For example,
$$
                   (D-2)^{2}(D-1)^{3}g = 0
$$
has solutions
$$
              g=(a_{0}e^{2t}+a_{1}te^{2t})+(b_{0}e^{t}+b_{1}te^{t}+b_{2}t^{2}e^{t}).
$$
This is because
$$
\begin{align}
   (D-2)^{2}(D-1)^{3}g & =
 (D-2)^{2}(D-1)^{3}(b_{0}e^{t}+b_{1}te^{t}+b_{2}t^{2}e^{t}) \\
    & + \;(D-1)^{3}(D-2)^{2}(a_{0}e^{2t}+a_{1}te^{2t})=0.
\end{align}
$$
You can see how this method generalizes to any number of factors.
Interestingly enough, this operator method is the first method used to solve the general equations. Heaviside came up with most of these methods, and used the $D$ notation, too.
One of the reason you don't read much about Heaviside in Math texts is that Mathematicians of Heaviside's time despised Heaviside because he could solve all of these problems using his strange operator formalism, and he mocked them when they questioned his methods. He used partial fractions with $D$, expanded in powers of $D$, etc. We can now make a lot of sense of what he did, but no so clearly then. In one of his articles he pondered the question: If a Mathematician is good, when he dies does he go to Oxford to or Cambridge? Eventually Mathematicians named only the step function after Heaviside (probably to mock him) when, in fact, it is the impulse function (derivative of the step function) that Heaviside used to solve time problems for $t \ge 0$. Heaviside is considered the Father of Electrical Engineering. J. D. Jackson has a wonderful article about how the Dirac delta $\delta$ function was really Heaviside's derivative of the impulse function. Another important contribution of Heavside is putting differential operators in vector form, which he first did for Maxwell's equations. Imagine: Maxwell had written all of his original equations in component form! Heaviside is almost nowhere credited for div, grad, curl either, but he should be.
A: It is not entirely clear what you are asking. For second order ODEs 
$$
y''-2λy'+λ^2y=f(t)
$$
that has the double eigenvalue $λ$ you can factor any solution by splitting off the smooth exponential factor as in $y(t)=e^{λt}u(t)$ where in turn $u(t)=e^{-λt}y(t)$. Then $u$ satisfies the ODE 
$$
u''(t)=e^{-λt}f(t)
$$
and integrating twice gives $u(t)=A+Bt+g(t)$ so that finally 
$$
y(t)=Ae^{λt}+Bte^{λt}+g(t)e^{λt}.
$$
and the constants $A$ and $B$ are determined by the initial values $y_0$, $y'_0$.

For a system 
$$
\begin{bmatrix}
\dot x\\\dot y
\end{bmatrix}=
\begin{bmatrix}
λ&1\\0&λ
\end{bmatrix}
\begin{bmatrix}
x\\y
\end{bmatrix}
$$
the same approach works, set $x(t)=e^{λt}u(t)$, $y(t)=e^{λt}v(t)$, then the system reduces to
$$
\begin{bmatrix}
\dot u \\ \dot v
\end{bmatrix}
=
\begin{bmatrix}
0&1 \\ 0&0
\end{bmatrix}
\begin{bmatrix}
u\\v
\end{bmatrix}
$$
with the solutions $v(t)=B$ and $u(t)=A+Bt$, where the constants $A$ and $B$ are functions of the initial values $x_0, y_0$.
Any system with an eigenvalue of geometric multiplicity 1 and algebraic multiplicity 2 has the above matrix as Jordan normal form and thus a solution containing $te^{λt}$ terms.
