Missing solutions in differential equations

[alexander]

Ok, if the topic did not describe it well, it is because it's not a simple thing to ask, so here's explanation for the question.

 

Looking at certain differential equations (diff eq), it is not hard to find equilibrium solutions. However as practice shows, upon solving the diff eq, that solution may be missing. here's an example of what i am talking about:

[math]\frac{dy}{dt}=t^2y^2[/math]

right here, we can see that if y was to be 0, independent of value of t, dy/dt=0 (it is called an equilibrium solution)

 

so now let's solve the equation:

[math]\int{\frac{1}{y^2}dy}=\int{t^2dt}[/math]

[math]\frac{-1}{y}=\frac{t^3}{3}+C[/math]

[math]\frac{-1}{y}=\frac{t^3+C_1}{3}[/math] where [math]C_1=3C[/math]

[math]y=\frac{-3}{t^3+C_1}[/math]

but no matter the value of c, there is no way that y will ever equal to zero at all values of, and thus an equilibrium solution is lost

 

i have other examples if you need them, here's the question to you, all mighty gods of math inexplicableness. While it's not always a zero solution that is missing, from what i have seen and been lead to believe, it seems like it's always an equilibrium solution that is missing....

 

Question: are there any situations, cases, specifically diff eqs when solutions other then an equilibrium solution are missing? I don't know if you will be up to this job, but, respect to anyone who embarks on this task.

 

BTW i can not find math lit that deals with this, it may just be another hypography discovery :P

[alexander]

btw if you want another equation where it's not the 0 solution that is missing, solve this one:

[math]\frac{dy}{dt}=y^2-4[/math] at this point [math]y=^+_-2[/math] are equilibrium solutions

[Erasmus00]

In both cases, in order to integrate you divide by the right hand side of the equation. This implicitly assumes the right hand said is not zero. Hence, you lose the solutions that only exist when the right hand IS zero.

-Will

[alexander]

Props, Erasmus

 

I understand why you loose these equilibrium solutions, i figured that out in this situation, but does that absolutely mean that you can not loose solutions other then the equilibrium ones, that's why i need to use your guys magic math wizardness, my math skills are way too poor to even begin thinking about how i can approach proving or disproving, or even just showing that there can or can not be other solutions that may be lost in the process of integration... (ps, no its not a homework question, if you were pondering, it's a curiosity)

 

Thanks a lot :phones:

[Qfwfq]

There are so many types of differential equations, there is no universal method and there are kinds which can't be solved in terms of standard things. Questions of this type need to be asked for a given kind of equation, the one you ask is somewhat undefined because there's no universal distinction between those found by "the method" and "the missing" ones.

 

I once read the anecdote about someone, after attending a lesson in differential equations, exclaming that it isn't mathematics but botany.

[alexander]

ok, don't just think that i will leave it unanswered :)

I'm back on the case of loosing solutions due to process of differentiation

 

The question still remains, are there situations when solutions that independently of other variables would not cause a diff eq to be equal to zero, be lost in the process of differentiation due to the way you differentiate, for example by the process of defining functions that are undefined at certain values throughout the differentiation process? There seems to be a minor debate between math professors that i have talked to about this topic, and they seem to think that there are other solutions that may get lost, but i have yet to think or see an equation that proves it, nor has anyone claimed to have such an equation, or a theorem as to why that can not happen...

 

Come on, this is fun, i hate math and this is something i'm pondering, i call on you math wizards (and i mean this in a good way) to dedicate a few brain cycles, you may see the light that is omitting me :eek2: