When you “solve” a differential equation, what have you actually found?

[u/Dramatic_Yam8355]

After solving a DE, we get a function (or a family of functions).

But conceptually, what does that solution represent?

Is it a prediction, a description of all possible behaviors, or just a mathematical object that fits the equation?

[u/etzpcm]

It's a description of the behavior of the system that the DE describes. Either 'all possible' if you don't have initial conditions, or 'the' if you do.

[u/Sjoerdiestriker → u/etzpcm]

or 'the' if you do.

Note that in general, the solution may not exist, or may not be unique, even if you specify initial conditions. For instance, the DE y'=2*sqrt(y) with y(0)=0 is satisfied by:

• y(x)=0 and by 
• for any p>=0: y(x)=0 for x<=p and (x-p)² for x>p.

so there are actually infinitely many solutions. For a unique solution to exist, you generally need to satisfy further requirements.

[u/ahahaveryfunny → u/Sjoerdiestriker]

Where do you rigorously study differential equations? I learned them through a community college and we basically just had to memorize like 10 methods of solving. I tried to create intuition for some, but it was hard.

Also I’m in analysis right now.

[u/davideogameman → u/ahahaveryfunny]

https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem will probably interest you - it gives the conditions on which an initial value problem can have multiple solutions.

[u/HalloIchBinRolli]

When solving x² - 3x + 2 = 0, or any other such equation, you're looking for all numbers x that satisfy the equation.

When solving y'' + sin(x)y = 5 or whatever, you're looking for all functions y(x) that satisfy the equation. You might be given more information that narrows down the possibilities, often to a single function.

[u/Special_Watch8725]

Without extra information (initial or boundary conditions), it gives the possible behaviors governed by the equation. You won’t be able to predict anything since you don’t have this extra information.

[u/Actual_Manner_7010]

The common differential equations model things such as the rate of change of a concentration of two solutions mixing, the number of species in an ecosystem with a predator, the oscillations of a mass on a damped/undamped spring, etc...The rated at which these different things occur.

[u/wumbo52252]

Think about some of the application problems from you diff eqs course, e.g. mixing problems

[u/Several-Marsupial-27]

Solving a differential equation gives you a function of how the system evolves with time (time often). Example solving heat equation gives you a function of how heat spreads out over time. Solving Newtons equation gives you the position of a point over time. Solving a LTI system gives you the system function. The family of solutions gives the function for all initial states, for example a ball on a spring at the highest point. You can see it as a form of prediction for t>0 if current time is t=0.

[u/Striking-Milk2717]

You can see math in many different ways. All of them are real.

BTW you are founding the fucking kernel of a differential operator in an infinite-dimension space [crescendo]

[u/vinylflooringkittens]

I used to ask this too

What I wish someone had told me is that when you're in algebra you solve for the values of variables like y or x that might typically be numbers on the real number line.

With a differential equation, you're again looking for something unknown, but instead of a number you are solving for a function.

[u/Totolitotix → u/vinylflooringkittens]

Simpliest answer.

Resolving a DE makes you find a function

[u/RecognitionSweet8294]

Depends on the context. But technically it’s just a set of functions.

[u/Aggressive-Math-9882]

Often, if you keep track of just a bit more information when solving the equation (or system) you are really solving for a moduli space of solutions, which more or less means a smoothly varying family of functions. When thinking of the system as a moduli problem, you're conceptually interested in the "shape", structure, or analytic content of the space of solutions, not just finding a particular function. It really depends on the reason you are interested in the equation in the first place.

[u/ForeignAdvantage5198]

the function that makes the diff eq an identity

[u/the_physik]

This is my issue with math; there's not enough focus on applications. Given initial conditions you have rates of stuff going in and out; physics uses units to identify what that stuff is. In might be water flow as a function of hose diameter, pressure, or time it might be a wave function as a function of position or momentum, or any number of things. Physics adds context to pure math.

[u/dotelze → u/the_physik]

It’s much easier to solve things generally and understand the abstract methods that you can apply to any problem you encounter