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/sci/ - Science & Math

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5163913 No.5163913 [Reply] [Original]

I'm working with a system of differential equations atm, and just want to make sure I don't do anything stupid while doing derivations. Google results weren't particularly helpful, so I figured I'd just ask here.

Suppose I have something in the form of <span class="math">\dot{x}(t) + a(t)x(t) = b(t)[/spoiler], then one can derive that <span class="math">x(t) = I(t)^{-1} \int I(t)b(t)dt[/spoiler] where <span class="math">I(t) = e^{\int a(t)dt}[/spoiler].
Is this still a valid formula to use if we have something like <span class="math">b(t) = Ae^{at} + e^{b-c)t[/spoiler] or do I have to do something different then? I don't see why it wouldn't work, but I'd rather not spend ages working things out that aren't correct anyway.

>> No.5163918

>>5163913

Bah, obligatory LaTeX fuck-up.

It was supposed to say <span class="math">b(t) = Ae^{at} + Be^{(b-c)t} + C[/spoiler]

>> No.5164446

Bump. Anyone?

>> No.5164493

No idea, but eigenfunction are cool

>> No.5164495

>>5163918
no
are you doing LDV models or something?

>> No.5164512

>>5163913
>system of differential equations
>system
>No matrices/vectors in your question
?

Also no, the formula is only valid for a first order eq. of the form posted. For a system of first order DE's you need to find the eigenvalues of your system then solve the eq. for the solution

x(t) = vexp(λt)
where x and v are vectors

>> No.5164597

>>5164495

It's an Optimal Control problem.

>>5164512

Well, I've got a system of equations and the variables within it are interconnected, but due to one of the equations being solvable really easily and through the FOCs of my Hamiltonian, I can determine the time paths for the remaining variables one by one by substituting already-derived time paths into other equations. In the end I'm left with some (rather long) equation in the form of <span class="math">\dot{x}(t) + a(t)x(t) = b(t)[/spoiler] where x(t) is not a vector, a(t) is rather simple, but b(t) is something like <span class="math">b(t) = Ae^{at} + Be^{b-c}t + C[/spoiler] (much more complicated, of course, but similar in structure). Finding eigenvalues for the system would be a pretty massive clusterfuck, so the clumsy method I'm using is sadly a lot easier to do.

Effectively, I'm wondering whether applying the afore given formula is valid for all forms of b(t), or whether there's an alternative method for when there are exponential functions within the b(t) (i.e., I'm simply not sure how general the formula is).

Pardon if all this sounds silly or trivial; I'm simply not particularly experienced with Optimal Control problems.

>> No.5164642

>>5164597
>Effectively, I'm wondering whether applying the afore given formula is valid for all forms of b(t),
Yeah, the only requirement is the form x'(t)+a(t)x(t)=b(t), a(t) and b(t) can be any function of t as long as there are is no x(t) (a(t,x(t)) etc.).
>alternative method for when there are exponential functions within the b(t)
b(t) can be an exponential function, but you could use the general method for linear eq. and solve for the homogenous eq. and then find the particular solution.

>> No.5164681

>>5164642
>you could use the general method for linear eq. and solve for the homogenous eq. and then find the particular solution.
Wait no, that would be insanely complicated. OP I think the formula is the lesser of 2 evils here.

>> No.5164713

Lost a year in uni due to sickness. Still haven't had more than a basic introduciton to differential equations and really really want to get there. ;_;

>> No.5164752

>>5164713
>Lost a year in uni due to sickness.
>a year
Damn...what was wrong?

>> No.5164935

>>5164752
Told them I had glandular fever, but honestly I had severe depression, thrown myself in front of a car, got arrested by the police, studied at home for a while and now after a few dramatic life changes (among which choosing a calm life of solitude over a social life full of drama, is the most important) I'm feeling much better.
(No, I'm not attention whoring. You asked.)

>> No.5164978

>>5164935
>glandular fever
I actually had this for real, but still went to uni because I've had it for most of my life, the only thing that makes cancer worse is that you don't die after it wasted years of your life.
>depression
Starting to become depressed too, I just don't see a future for me anymore, but that seems to be a general trend among STEM majors.
>I'm not attention whoring
This is /sci/, attention whoring is obligatory.

>> No.5165003

>>5164978
Glandular fever can indeed be very serious in some cases. That's why I used it as an excuse.
--
My engineering degree, related projects I'm working on and mathematics in general are actually my major love in life. I really love doing it.
--
I disagree. This board isn't meant for complaining about your personal life (not that I was even complaining thought) so if you don't mind I'm going to stop talking about it now.

>> No.5165066

>>5164935

Science is the reason why most of us haven't killed ourselves yet.

Without science there wouldn't be a depth to life it would just be like watching the world through a tv screen...

>> No.5165079

>>5165066
>Without science there wouldn't be a depth to life
Couldn't agree more.

>> No.5165099

>>5165079
Nothing would have depth without the 3rd dimension. 3rd dimension>science

>> No.5165107

>>5165099

>implying im not a 4th dimensional supreme omnipotent immortal entity

>> No.5165119

>>5164642

Alright, awesome. Thanks!

>>5165079
If you define 'science' as any form of satisfying curiosity, then I agree as well.

>> No.5165147

>>5165119

>implying satisfying curiosity can exist outside STEM