What does this statement regarding auxillary equations mean?

"e^(alpha*x) and xe^(alpha*x) are independent solutions of the differential equation when the auxiallry equation has repaeted roots".

To put it in a form you may be more familiar with, the solution becomes (A+Bx)e^(alphax).

"e^(alpha*x) and xe^(alpha*x) are independent solutions of the differential equation when the auxiallry equation has repaeted roots".

A second order DE like $\frac{d^2 y}{dx^2} + 2\frac{dy}{dx}+3=0$ will always have two, *independent* solutions. This can be proven using techniques not available at A level.

Here "independent" essentially means that one solution is not simply a multiple of the other. So e.g. if you found that $y=e^{2x}$ is a solution of some second order DE, it would be no good to claim that $y=2e^{2x}$ is another solution, since that's just a multiple of the first - it's not independent of the first.

If you have a DE whose auxiliary equation has two distinct roots, say 2 and 3, then the independent solutions of the DE are $y_1=e^{2x}$ and $y_2=e^{3x}$, and the most general solution is a so-called "linear combination" of both i.e it's $y=c_1 e^{2x} +c_2e^{3x}$ where $c_1, c_2$ are any old numbers you like. (It doesn't matter what you choose; any $c_1, c_2$ will give you a solution - but if you have initial conditions to satisfy too e.g. $y(0)=1, y'(0)=2$ or something, then only one specific $c_1, c_2$ pair will work)

Note that $y_1=e^{2x}$ and $y_2=e^{3x}$ are indeed independent - you can't turn $e^{2x}$ into $e^{3x}$ by multiplying it by any fixed number. (Try to figure out why).

However, if you have a DE whose auxiliary equation has repeated roots, then you don't have two independent solutions - you will end up with, say, $y_1=e^{2x}$ and $y_2=e^{2x}$.

But all is not lost: it turns out that some more advanced maths saves us: there is something called Abel's formula, that relates the independent solutions of a DE. Given Abel's formula and one solution of a DE (in this case $y_1=e^{2x}$), you can write down an equation that $y_1,y_2$ must satisfy, and solve it to find $y_2$.

In the case where the auxiliary equation gives us $y_1=y_2=e^{ax}$, Abel's formula shows us that, in fact, we must have $y_2=xe^{ax}$. But at A level, no one tells you this: you are merely told that for repeated roots, the two independent solutions have the form $y_1=e^{ax}, y_2=xe^{ax}$ and you just have to take it on trust.

(For those interested, Abel's formula is a statement about the Wronskian of the solutions, and for repeated roots, we have that the solutions must satisfy:

$W(e^{ax}, y_2)=e^{2ax}$

where $W(y_1,y_2)$ is the Wronskian - Google is your friend for details)

very informative Sir !