Basis of second order Differential equations

[brisk timber]

Why is it necessary for n order derivatives to have a basis of size n. And if you have repeated roots why do we just add an x coefficient. Doing the calculation it seems to always work but why?

[tender otter]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[modern mirage]

The characteristic polynomial of the differential operator defining the DE will be degree n, so by fundamental theorem of calculus it has n roots counted with multiplicity

you add the factors of x for repeated roots to ensure the fundamental system is independent

[brisk timber]

Sorry thats a lot of technical language

What do we mean by characteristic polynomial

[modern mirage]

the function of r you get when you ansatz y=e^(rt)

[brisk timber]

Oh right I see yes

[modern mirage]

assuming constant coefficients, you have some $L[y]:=\sum_{i=0}^na_i\dv[i]{y}{x}=0$

[brisk timber]

Yes the first answer now makes sense

[tender rapids]

Omegabet_

[brisk timber]

So for the repeated roots part?

Fundermenal system is inderpendant?

[modern mirage]

L is a linear operator, so the solutions to L[y]=0 is a linear subspace

hence, by the 1st part, it is finite dimensional, specifically n dimensional

[brisk timber]

So I should be thinking of the differential operator in matrix form?

[modern mirage]

the exponentials form a basis by construction, but if you have a repeated root, then that list isnt n elements large

since ${...,e^{rt},e^{rt},...}$ will remove one of the listings, which wont be n-elements large

[tender rapids]

Omegabet_

[brisk timber]

Basis by construction?

[modern mirage]

but you can easily check that ${e^{rt},te^{rt}}$ is independent

[tender rapids]

Omegabet_

[modern mirage]

what do you think the Ansatz does?

they form the basis

[brisk timber]

Sorry what’s an ansatz

[modern mirage]

lucky guess in German

you ansatz $y=e^{rt}$ to derive the char poly

[tender rapids]

Omegabet_

[modern mirage]

$L[e^{rt}]=0\to e^{rt}\sum_{i=0}^n a_ir^i=0$

[tender rapids]

Omegabet_

[brisk timber]

Ah right where we derive the polynomial from

[modern mirage]

so, since L is linear, and by construction you find r that makes those exponentials solutions

any linear combination of them is a solution

[brisk timber]

Yes yes okay

[modern mirage]

so the solution space is $\operatorname{span}(e^{r_1t},...,e^{r_nt})$

[tender rapids]

Omegabet_

[brisk timber]

What’s the assumption that our our solution space has such a large span in the case of repeated roots

[modern mirage]

so you want the list of exponentials to be a basis and preserve the fact dim(solution space) is n

if you have repeated roots, say r_1=r_2 wlog, then the list in the span is only n-1 elements

which cant happen cause then the roots werent counted correctly

so you need a way to include both instances while preserving the independence of the set

[brisk timber]

So we need n solutions

[brisk timber]

We know this is inderpendant

What then makes that a solution

Where if it’s repeated twice we can have an x and twice we can have an x^2?

Or rather t in your case

[modern mirage]

yes

if a roots is repeated 3 times, then you need 3 instances of it in the spanning set while making the set independent

so you get $e^{r_1t},te^{r_1t},t^2e^{r_1t}$

[tender rapids]

Omegabet_

[brisk timber]

Okay but we could make them inderpendant by having $e^{r_1t},t^2e^{r_1t},t^3e^{r_1t}$

[tender rapids]

CBrocks294

[brisk timber]

But this wouldn’t work as a solution would it

Is there something special about using the lowest power of t and going upwards

[modern mirage]

simple is better

but you can re-derive it more formally by guessing that it's some $v(t)e^{r_1t}$

[tender rapids]

Omegabet_

[modern mirage]

then depending on the order determines what v is

you'll get polynomials that increase in degree as the multiplicity of r_1 changes

[brisk timber]

Alright I’ll have a little go at deriving it tommorow then thank you very much @modern mirage

[coarse birch]

@brisk timber has given 1 rep to @modern mirage

[modern mirage]

example from my textbook

[brisk timber]

Oh wow yes thank you that’s very clear

It’s necessary for the second dif to be zero

Hence specifying any constant and any x term

[brisk timber]

+close