>>17087360Um... 4?
>>17087361y = 4? for all x?0 = 12 + x ln(4)which does not hold for no x apart from -12/ln(4) so no. Im asking a general solution.
>>17087360It all depends on the initial value.If y(0) <= 0 then it doesn't have a real solution.If y(0) >= 1 then y(x) will grow rapidly (slightly faster than e^(3x))
let z = ln(y) s.t. dz/dx = z' = y'/y z' = 3 + x z exp(-z)let a = -z s.t. da/dx = a' = -z' and W(a) := a exp(a) as lambert Wa' = x W(a) - 3analyze the Green's function solution to a' - x W(a) = 0This is a trivial separation of variables wherebyda/W(a) = x dx which itself permits trivial, well known solutionsEi(W(a)) + exp(W(a)) = x^2/2 + C where C is a constant of integration and Ei is the exponential integral. replace back to original variable y whereby a = -z = -ln(y)Ei(W(-ln(y))) + exp(W(-ln(y))) = x^2/2 + Cq.e.d.
>>17087375This seems to not work according to gpt pro. since x = W(x)e^W(x) a(x) = W(a(x)) e^W(a(x)) maybe it didn't get something but the idea is interesting.
You are all retarded. OP baited you all into doing his homework with THE laziest bait.
>>17087386Im OP. I made up this problem to showcase how vulnarable our theory is to slight deviations from known to unkown equations like:[eqn]y' = 3y +x y^2 [/eqn] bernoulli type [eqn]y' = 3y +x \ln y[/eqn] this. But in fact I was studying to my midterms lol this is somewhat a right comment.
>>17087390This is not on you OP. It's the retards in your thread that have shattered my confidence.
>>17087404Im OP.I suppose I understand
>>17087360let me see if I can remember the math lectures I never attended a decade agothe troublesome term obviously is x * ln(y) making this ODE is inseparable. Tried a couple of substitutions then gave up. I expect this only has a closed form within 0 <= y <= 2, where the power series for ln(y) converges. Maybe not even then, I didn't check. Outside of that i dont know
>>17087428EDIT: nevermind, I found the solution
>>17087434>EDIT: nevermind, I found the solutionwhats it tho
Define "solve".
>>17087463Find all initial values [math]y(0)[/math] for which the ODE has a solution defined for all [math]x \geq 0 [/math]. If the initial value is too close to zero then the solution will eventually hit the x-axis and stop there.
>>17087360Just substitute u=ln(y), then probably solve with W
>>17087511Claude says this doesn't work
>>17087360Just solve it numerically
>>17087365Could you double check?
>>17087379gpt pro is wrong.
>>17087360if you really wanted to cry about this, the vast majority of integrals don't have a good closed-form solution either. it's only for a certain class of DEs that easy techniques work
>>17087538It may be wrong. I don't wanna trust gpt as well but I became too dependent on it.>>17087629Yeah... I wish everything had a closed form solution
>>17087726chatgpt got confused because of separation of variables. a(x) is a function of x. but we also don't care. integrate w.r.t. a. like when you have dy/dx = yx. separate to dy/y = x dx. integrate. it doesn't matter that y(x) is a function of x.
>>17087361kek
>>17087360>>17087390I will get stoned for this but that’s not a differential equation because ln(x) is not a function. It’s not a function in any meaningful sense because you can’t have it assign in general an element from an input space to an output. ln(x) is the name we coined a numerical solution to the differential equation y’=1/x, but a numerical solution is not a function and I’m tired of pretending otherwise. You’ve essentially hid an unsolvable differential equation inside another one.
>>17089790>you can’t have it assign in general an element from an input space to an output???ln(x) is a bijection when x is a positive real number.Are you also going to say that exp(x) or e^x is "not a function" because it's just a numerical solution to the differential equation y'=y ?
>>17089790saying it’s a bijection doesn’t provide you with a magic wand. there is no way to map a number to ln(x), it’s interminable. yes, same for e^x. you can get away with finite polynomials for everything btw and in some sense, it’s exactly what we do irl. it’s hard to digest.
>>17089846>e^x is not a functionOkay anonDo you also think pi is not real because there is no way to determine pi?
>>17089952certainly not real in the physical sense, perfect lines and circles don’t exist in nature. there is also no place the universe could store such a number or compute such a number. at best pi “exists” in a platonic sense. sure it’s useful, but that alone isn’t a criterion for a foundation of math. the ends shouldn’t justify the means imo. tell me why you need pi, or e^x again, maybe you can change my mind.
>>17089995>perfect lines and circles don't exist in natureProve it. Prove they truly don't exist, as opposed to our inability to measure them.
>>17089996The Bekenstein bound prescribes a limit on the information that can be stored by physical objects. The universe we live in prohibits the existence of perfect lines and circles
>>17089996false dichotomy. there's functionally no difference between the two. to our current knowledge, they do not exist. despite looking vast and wide and hard
>>17089790Well lnx assigns exactly one output to every input in its specified domain which is to be positive numbers?
>>17090003>Bekenstein boundliterally has pi in it
>>17090126it’s a bound, it doesn’t determine S.>>17090124you postulated what it is, but you don’t say how it makes the assignments.
>>17090155It's trivial you can understand it by your own
>>17090155>it’s a bound, it doesn’t determine S.when you say "it" what are you referring to? If you're referring to a hypothetical bekenstein bound, such a thing doesn't exist
>>17090170i never said that pi, ln, etc don’t “exist”. if you look above, i wrote that they may very well exist in a platonic sense, and this is in contrast with the physical world. i see no contradiction that a physical limit contains transcendental quantities, by virtue of the real world not trespassing this limit, which here is an ideal mathematical constraint on physical reality>>17090158>triviallol>left as an exercise to the readert-thanks
>>17090180Does any function exist in the physical world?
>>17090207the ones that can be computed
>>17087360I hate diff eqs so much. They do look nice to write out on paper and solve thoughbeit
>>17090227Example?
>>17090238polynomials
>>17090260Where does the polynomial f(x) = x^3 exist anywhere in the universe?
>>17090264inside the cpu via electric signals
>>17090227>>17090260Proof that we can compute polynomials? How do you know our finite representation isn't uncomputable on the inside? Like what if 3^3 = 9 is just a "representation" of a number that's uncomputable on the inside, just inaccessible to us?
>>17089790Skill issue.That is like saying f(n) = n / 2 is not a function because doesn't have an integer solution when x is odd.
>>17090155>you postulated what it is, but you don’t say how it makes the assignments.That is not required in standard axiomatic mathematics. Only constructivists would ask you to do so in order for your reasoning to be correct.
>>17090269You’re asking the right questions and I’m not claiming I have a satisfying answer. I want you to consider though that you’re questioning the validity of finite algorithms that don’t even care what a number is, since algebra is not concerned with this. Now ask these same questions with the same rigour for notions that make claims about infinities.
>>17090277>constructivistsbased
>>17090282>questioning the validity of finite algorithmsYou can do symbolic computation and treat pi just like 3 or 5 if you so wish. The same applies for any real number you can think of.
>>17087360we can't?[math]x = (y' - 3y)/ln(y)[/math]then y can't be zero (0), which means this equation has no solution!
>>17090294sure but even then, it doesn’t eliminate the fact that that the reals are uncountable and the number of algorithms is countable. this should give an indication about the nature of the continuum imo
>>17090337What does this have to do with >>17089790 though? The inputs to a Turing machine are uncountable btw (inb4 something retarded like "Turing machines don't exist")
>>17090344Turing machines are a type like computing and concepts in general. Substitutional grammars dont necessarily justify the operations and behaviors of their production. Inventing a category of thing that can perform some operation on an infinite string isnt really a problem if its decoupled from grammar, math, or computing. Irrational numbers are a operation on infinite decimals for instance, identity minimally but also others, and they do have some coherence. So can you highlight what you think the controversial aspect of your post is?
>>17090344My point is that symbolic computation doesn't say anything about existence over the reals. There are only countably many algorithms, whereas the reals are uncountable, so the computable reals form an incomplete, countable subset of the continuum. Classical existence theorems for ODEs (eg. Picard–Lindelof) rely on completeness, and their conclusions don't automatically transfer to that restricted setting. So what exactly are we asserting when we say we've solved the differential equation in OP?
>>17090413Since we can write down an expression for the function, we know that f(x) is computable when x is computable. Despite this, f(y) still exists for uncomputable y. It is in this sense (taking the uncomputable and computable reals) that we say f solves the differential equation. And of course, we couldn't imagine an uncomputable y to input to f, so for any use at all, our input x will be computable, and so will the result of the function f(x).Since the computable reals aren't complete, it's ill-defined to even talk about a differential equation in that domain.
I have never needed any of this shit after getting a job and touching grass and having sex thank god AI will get rid of useless labor like this
>>17087360>weLol okay buddy
>>17087365>Im asking a general solutionlist the numbers you are interested in knowing
>>17090324>y can't be zeroHow do you know? did you solve it and express y in terms of x and plug x = 0? Even if it can't take 0 zero then its domain is real numbers excluding zero and it becomes well defined I suppose.
>>17090618>its domain is real numbersare the real numbers in the room with us right now?
>>17090429>Since the computable reals aren't complete, it's ill-defined to even talk about a differential equation in that domain.I think I agree with this although I don’t know how to interpret your conclusion.1) Are you saying that we should enlarge our definition of what a number is to include the uncomputable reals to accommodate the notion of a differential equation?2) Or does your claim go in the opposite direction, meaning that differential equations are the wrong approach and we take the computable reals as the building blocks of analysis?3) Or I misunderstood you completely and you’re arguing for something else.
>>17087470>allYou really need all?Not just some subset that's relevant to an actual physical situation?Rude
>>17090766Uncomputable reals are numbers, yes.
