/sci/ - Science & Math

[Anonymous No.15324948]

>study ordinary differential equations
>homogeneous ones make absolute sense, you can intuitively reach the conclusion that it's a linear combination of exponentials just by knowing about linear algebra and calculus
>arrive at non-homogenous equations
>suddenly all intuition disappears, the professor just says "suppose the solution is in this form" and calls it the "variation of parameters method"
bros, is there any actual logical theory that justifies this shit? like, is there any actual step by step reasoning to move from homogeneous to non-homogeneous? is this "I dreamt it last bight" shit really the only way we could solve these equations?

[Anonymous No.15324984]

>>15324948
There is no algorithm for solving non-homogeneous differential equations. You have to guess and check. Certain classes of equations have obvious solution forms but the general case is unsolvable

[Anonymous No.15324985]

>>15324984
I forgot to say they are constant coefficients. Does that help?
Is there a proof that they're unsolvable?

[Anonymous No.15325038]

>>15324985
don't know, I'm not mathematician

[Anonymous No.15325039]

Filtered, talentless and so on...

[Anonymous No.15325041]

Lol welcome to the real world. You didn't expect math to be that easy did you?

[Anonymous No.15325042]

>>15325039
lol, the typical reply of someone who has nothing to say

[Anonymous No.15325043]

>>15325041
>is there a logical connection?
>LOL MATH IS HARD LOL KEKE LMAO ROFL
I bet you can't even explain it

[Anonymous No.15325058]

>>15325043
Real stuff is non-linear. The homogeneous stuff is an assumption made for easier algebraic analysis.

[Anonymous No.15325060]

>>15325058
Wow another non-answer.
Yes big guy, the real stuff is non-linear, now explain how to get from homogeneous to non homogeneous without oracle-tier logical jumps

[Anonymous No.15325076]

>>15324948
Method of integration factors is an equivalent method, but a lot more straightforward. My book showed both methods and I dont understand the reasoning behind VoP.

[Anonymous No.15325080]

>>15325060
There's no how to get. One is a simple abstraction, the other is a more complex abstraction. Homogeneous odes can be thought of as dynamical systems with no input, non-homogeneous odes have some input that drives the dynamics. Your need to get somewhere is what is driving your misunderstanding.

Think of a pendulum swinging in a vacuum vs a pendulum that has to contend with air drag on earth. One has no input and will have the same dynamics. That is the amplitude, the duration of the swing will produce a near perfect sine wave. The pendulum always swings to the same end and at the same time. The other non-homogenous one will decrease in amplitude and phase. That is it will swing slower and reduce the angle over which it extends. You can also provide an input that allows it to have a larger swing, etc.

[Anonymous No.15325085]

>>15325080
Nigger, do you fucking understand my question?
I am asking how to derive the non-homogeneous solution from the homogeneous one WITHOUT LEAPS OF LOGIC.
I am not asking you to explain what the fuck it means, I already know that

[Anonymous No.15325088]

>>15324948
The general solution for linear non-homogenous differential equation is pretty simple, I'm sure you know it already but for a general differential equations there isn't a general solution, it's a case by case thing.

[Anonymous No.15325101]

>>15325085
You are a retard if that's all you got. I just told you that one has an input and the other doesn't (or you can think of it as the derivative of an equation y w.r.t t plus some other function of x(t) being equal to a function that doesn't depend on time (depends on x alone) vs one that evolves with time(depends on t)) . If you already knew that and can't extrapolate then I can't help your retarded ass.

[Jealous No.15325116]

>>15324948
>bros, is there any actual logical theory that justifies this shit?
Looking up the solution in a table of solutions is how everyone solves ODEs and PDEs. After Schrodinger found the Schrodinger equation, he had to find the solution to the hydrogen atom in someone else's book. He didn't solve it himself. The main advantage of taking undergrad ODE is that you get familiar enough with the equations to know how to look up the solution to one if you needed to. Now there's Mathematica but there didn't used to be, and there are Luddites like myself who have never taken the time to learn it. I really ought to do it some time.

[Anonymous No.15325130]

>>15325085
If you knew what it meant for one to be homogeneous and the other not to, then you wouldn't need to ask how to derive one from the other.

First understand what an non homogeneous ode is. You have a function of x(t), y(x(t)). Find its derivative w.r.t t and add some other general function of x(t). That equation equals another function that depends on t. That is, it evolves with time. A homogeneous ode on the other hand is the derivative of y(x(t)) w.r.t t plus some other function z(x) all that being equal to a function of x that doesn't change with time.

There's no general way of deriving the solution x(t) of the former from the latter because the latter doesn't depend on time.

So you have to plug in different x(t)s depending on how they depend on t. You can't plug in an algebraic x(t) if the non-homogeneous ode equals a logarithmic or trigonometric function.

[Anonymous No.15325135]

Solving equations is a spook. Study qualitative theory of ODEs

[Anonymous No.15325144]

>>15325130
>There's no general way of deriving the solution x(t) of the former from the latter because the latter doesn't depend on time.
You could have simply said this. So I was right, you can only solve them through magic substitutions. I thought this was only true for more complex types of DEs

[Anonymous No.15325466]

>>15324985
Just do a fucking Laplace transform retard.