I look down at the Ar15 I am holding with night vision sights, a bump stock, bipod, and a 90 round drum magazine

"Well, I can feel comfortable knowing I WOULD NEVER let machines think for me! I am not an idiot sheep like that."

I pat the assault rifle oblviously as if it were a beloved pet cat

Reality check, at school there were way more average people than kids at the top of the class. These arguments are made by average kids, and a society that says being intellectual and smart is a desirable trait

How does reading socrates and plato equate to letting a machine do the thinking for you, this article is basically saying that when you go to school and talk to a teacher and learn to ask questions and sift through right and wrong answers, somehow, deprives people of learning...this fallacy is so annoying and its made by people who dont understand how learning is done

Of course the pure math guy makes the Saruman reference.

People do things because they like them. Sci-Fi authors have been warning us about letting our machines think for us for more than 100 years.

Betteridge's Law of Headlines: no.

What a bunch of nerds.

(Like in a good way)

As someone who was recently running into some of these exact limitations when designing something in CAD with cubic bezier curves, this is selling me

Sorry, I was being a bit sloppy in the first answer. Your f=A•t•exp(b•t) term is, in general, part of the expected function space. (I just kinda forgot because in control systems, we often assume distinct real system poles for simplicity in teaching the material.)

Hi again ! In your reply, you told me :

this means f is a sum of exponetial

However. For the case wher f is of the form f=A•t•exp(b•t), i don't know how to convert that as a sum of exponential... I'm asking because i feel like i could just take x to be an indefinite sum of exponentials which might end up giving me some quite nice solutions... If only i could get a sum of exponentials on the other side of the equation ;)

Damn, well i have my work cut our for me :D once again, thank you for your helpfulness.

Hmm...this feels like a Lyapunov-type problem, but there's some issues. The vector function [x2,x1^-4 f] has no equilibrium, except when f = 0. So assume that f = 0. Then, the system is [x1',x2'] = [x2,0] = [0 1;0 0][x1,x2]. I believe that the zero solution satisfies your original differential equation. You would then need to find a Lyapunov function that guarantees the system is globally asymptotically stable around the zero solution. (Make sure to check for "globally asymptotically" in the stability theorem, e.g. in Theorem 3.2 of Haddad and Chellaboina (LibGen). The "asymptotic" part is what gives you the steady-state result you're looking for.) By taking the second component, I would then conclude that the limit of X' is the limit of x2, which would be 0.

But that's assuming zero input and initial state, and Lyapunov functions can be hard to find. Linearizing fails; you would have to guess a Lyapunov function.

Honestly, from an engineering perspective, I think you should just simulate the system. In particular, instead of leaving f(t) as a function, I suggest plugging in the second-order ODE and numerically solving the resulting composite fourth-order ODE system for your RLC values and some deviations from those values. Simulating it might give you an ansatz (educated guess) on if or where the system has equilibria.

The limit of x' as t goes to infty would be good enough, but if i could get x'(t), it would allow me to take into account partial capacitor discharge which would be a much better fit.

Thanks you so much !! As for the 4th power, it comes from the fact that the magnetic field on the central axis of a coil is proportionnal to 1 over the cubed distance to the center of the coil. Since the force applied to the magnet is proportional to the spacial gradient of the magnetic fueld at that point, you have to differentiate the previous expression which give some x^-4.

Also, what specific further analysis do you want to do for the differential equation? Because you might not need the solution for it. Like if you take a class on nonlinear control or dynamical systems, 90% of the class is basically "how do we do anything but solve the differential equation analytically?".

The fact that this comes from a physicall model means that there is a solution, no ?

Good instinct, but nope! There might be some phenomenon your model is neglecting where the neglected phenomenon allows the solution to exist but the model fails to have a unique solution. It also could have a non-unique solution (i.e. many solutions).

Do you have a link to those equivalent system, i can't find it online and anything i can search up on is nice.

So the first move is to simply solve the equation for X'':

X'' = X^-4 f

Unfortunately, for the next technique, it's typically buried in any nonlinear control book. I'll give the technique below. I'm gonna do it for a fourth order system to make the technique clear, but it applies for ODE systems of any size.

Say you have the following fourth order scalar differential equation:

X''''(t) = F(t,X(t),X'(t),X''(t),X'''(t)) Make the following substitutions:

x1 = X
x2 = X'
x3 = X''
x4 = X'''

If you have initial conditions at some time t0, then they must be:

x1(t0) = X(t0)
x2(t0) = X'(t0)
x3(t0) = X''(t0)
x4(t0) = X'''(t0)

(In your problem, t0 = 0, plus your problem is time-invariant so it doesn't matter, but I'm describing the technique here for time-varying problems for generality, where it does matter.)

Differentiate the four equations as follows:

x1' = X'' = x2
x2' = X'' = x3
x3' = X''' = x4
x4' = X'''' = F(t,X(t),X'(t),X''(t),X'''(t)) = F(t,x1,x2,x3,x4)

Stack these into a vector as follows:

[x1,x2,x3,x4]^T^ = [x2,x3,x4,F(t,x1,x2,x3,x4)]^T^

(The ^T^ just means "make it a column vector".) You now have a system of four coupled first-order ODEs that any ODE system solver will solve. By applying the technique described above to the second-order ODE X'' = X^-4 f, you get [x1,x2]^T^ = [x2,x1(t)^-4 f(t)]^T^.

This means that f is drawn from the space of solution ms of a second order homogenous liner differential equation (from an RLC circuit with no generator).

Got it, so the forcing is a sum of exponentials. So your input space is a subspace of the infinitely differentiable functions (except possibly at 0 if you choose to model the switching-on of the circuit as a step function). But I'm a little suspicious of the X^4^ term? Like I'm not a "magnets guy" although I've taken electromagnetic fields, but I don't think that's the type of nonlinearity I would expect to see (in particular , the fact that it's fourth power). I.e. if you derived this model yourself, double-check the derivation.

I would really like to be able to solve it analitically (further analysis of the results is required so the more mathematical the solution, the better.) I will have x(0) and x'(0). This question comes from trying to solve for magnet acceleration inside a coil. This means that f is drawn from the space of solution ms of a second order homogenous liner differential equation (from an RLC circuit with no generator).

The fact that this comes from a physicall model means that there is a solution, no ?

Do you have a link to those equivalent system, i can't find it online and anything i can search up on is nice.

Thanks so much for such a complete answer. :D

Control theory nerd here. I do a lot of work with differential equations, but I usually don't solve them exactly. I usually do other analyses like stability, controllability, observability, etc., where you don't need the closed-form solution to do your work. But I do know how to get numerical solutions.

Answer: depends:

Do you need to solve it? If you need to solve it, do you need an exact solution, or does a numerical approximation work? Do you have any initial or boundary conditions? Does this equation come from any physical system or model? What function space is f drawn from?

For proving existence and uniqueness (e.g. checking the conditions of Picard-Lindelöf) and deriving a numerical solution, my first step would be to convert it to the equivalent system x1'(t) = x2(t) and x2'(t) = ( x1(t)^-4^ ) f(t). (Here, you basically just add an extra state variable.) (We also get X(t) ≠ 0 unless f(t)=0.) Then, once I've checked that my initial conditions and driving function f will indeed produce a unique solution, use any numerical quadrature package to see what it looks like for different f and initial conditions, plugging in the right-hand sides of x1'(t) = x2(t) and x2'(t) = x1(t)^-4 f(t).

I could not personally derive a closed-form general solution of it as an initial value problem. I checked EqWorld's section on second order nonlinear ODEs and systems of two first-order ODEs, as well as my copy of Polyanin's handbook of exact solutions (LibGen) and I could not personally find a useful closed form solution. That doesn't mean it's not possibly in there, just that I'm not smart enough to find it.

I'm curious, how did you "fall upon" this differential equation?

Oh, ffs. No one is saying there aren't a ton of potentially positive uses for this technology. Obviously, there are.

The problem is that the capitalist class doesn't give a shit about using this technology to improve people's lives. They are overwhelmingly using it for purely profit-driven endeavors...none of which are going to actually help people. Every purpose is explicitly intended to control, manipulate and exploit the away parking class. They want to use it to strip us of our rights, our privacy, and out collective agency.

So, the question isn't, "What could we use this technology for?" It's, "What are they actually going to use this technology for?"

And the answers are not good, man.

I'm not sure how the plagiarism chatbot is going to do something new when it doesn't have a humans idea to rip off

Woah! I’m making the same amount of progress solving the toughest math problems as a multi-billion dollar AI!

Doubt it. I stick to the basic problems. Those all remain the same, but there could be updates in the advanced stuff i'm unaware of.

I did so many of these. Yes it's a excellent resource

Project Euler is the lifehack for job interview prep.

Find problems that are appropriate for your skill level, write out answers on paper or on a whiteboard, then once you're happy with them, put them into the computer and fix them up until they run. Keep doing that until you feel well-practiced with your ability to put code together without having an editor and think quickly about new problems you've never seen before. (For me it took a few days of practice.) Congratulations, you're now the guy or gal who can show up and eat the interview problems alive and look comfortable and confident doing it.

Yes

I did quite a lot of them maybe 10 years ago. Are they still being updated?