fundamental set

What?

now read again, it's edited

I would like to repeat the earlier "what?" Put some detail and context into your question, otherwise people won't be able to help you.

ok sure check again

$y_0 ^1 \cdot y_1 ^2 - y_0 ^2 \cdot y_1 ^1 \neq 0$

Then we have a fundamental set of solutions. Obviously we don't have a fundamental set if the values equal 0 but is there anything else i can say about the values or the equation they come from if the above is equal to 0?

Like repeated roots or something?

also do the subscript 0 and 1 mean to those powers?

Edit:With the subscript of $y^1\ and y^2$ being 0 and 1 are the powers of the terms $y^1\ and y^2$ also to the powers of 0 or 1 for example is the term$y_0 ^1$ to the power of 0 when we sub in $y^1$ into $y_0 ^1$. Or do i have the wrong idea and misunderstood what the terms mean?


And what does

$y_0 ^1 \cdot y_1 ^2 - y_0 ^2 \cdot y_1 ^1 \neq 0$
mean if the equality is actually equal to 0, is there anything else i should about this other than the fact that this does not form a fundamental set of solutions?

You really are not clarifying anything there. If you don't know whether the superscripts denote powers then you have to give us some context regarding what the variables mean. The only thing I might meaningfully offer is that it looks like a 2x2 determinant being nonzero - but that's just some educated guess.

Why are 0 and 1 used for subscripts?
Why are 1 and 2 used for superscripts?
What do these variables denote and where did the question come from?
What is a fundamental set? (I googled this and found nothing helpful)
Also what does the dot mean - is that multiplication or perhaps the scalar product?

is this something to do with DEs? As RichE said it looks like a det so maybe a wronskian? though what the exponents quite mean I'm unsure.

Edit2: In the section of math models.

Where a difference equation has a fundamental set if
$y_0 ^1 \cdot y_1 ^2 - y_0 ^2 \cdot y_1 ^1 \neq 0$

Where a fundamental set is a set of solutions which make up the general solution for example $a_n =A(5)^n +B(2)^n$ where 5 and 2 are part of a fundamental set only if they satisfy the formula(i think)

If you stick 5 and 2 into this $y_0 ^1 \cdot y_1 ^2 - y_0 ^2 \cdot y_1 ^1 \neq 0$ do you get (5)^0 x (2)^1 - (2)^0 x (5)^1 or do you not use 0 and 1

I'm sorry but this really isn't helping much. We don't have the same set of lecture notes in front of us as you do to make sense of this.

"In the section of math models." - that's rather vague/opaque and not helpful.

"Where a fundamental set is a set of solutions which make up the general solution for example $a_n =A(5)^n +B(2)^n$ where 5 and 2 are part of a fundamental set only if they satisfy the formula(i think)" - that very much doesn't say what a fundamental set is. Can you clarify? I do know how to some linear difference equations (and wrote a chapter of a book on this recently) but still can't make sense of this.

"If you stick 5 and 2 into this $y_0 ^1 \cdot y_1 ^2 - y_0 ^2 \cdot y_1 ^1 \neq 0$ do you get (5)^0 x (2)^1 - (2)^0 x (5)^1 or do you not use 0 and 1" - how can we answer this, not knowing what anything means?

I don't mean to be rude, but I have been on TSR for 12 years, frequently replying in the maths forum, and have never been so stumped/confused by a request for help.

ok
https://cdn.discordapp.com/attachments/322819040059850757/373843646136713217/unknown.png
https://cdn.discordapp.com/attachments/322819040059850757/373843688700248064/unknown.png
https://cdn.discordapp.com/attachments/322819040059850757/373843723156455434/unknown.png

Thanks - so the superscripts refer to choices of independent solutions ot the recurrence relation and not to powers.

The "preferred solutions" (1) and (2) are sequences of numbers and the subscripts then refer to the different terms of that sequence.

Let me further give an example which might clear things up.

Consider the solutions - each of which will be a sequence - to the recurrence relation

$y_{n+2} = y_{n+1} + y_{n}$

where n is a non-negative integer

One set of solutions are the Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, ...

An independent set of solutions is 1, 1, 2, 3, 5, 8, 13, ....

You might call the first $y^1$ and the second $y^2$.

Subscripts would be referring to terms in these two sequences.

And any sequence solving the recurence relations would be some linear combination of the two.

Right ok so we're talking the the 0th term of the 1st sequence, multiplied by the 1st term of the 2nd sequence minus the 0th term of the 2nd sequence multiplied by the 1st term of the first sequence, and as long as that isn't 0 we have a set of fundamental solutions?

Yes - that's a condition for the linear independence of the two sequences.