Symmetry in Differential Equations: Benefits & Consequences

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In summary, the symmetry of a system of simultaneous DEs does not help us find the solution faster. However, it does offer a similarity between the two functions which might make them easier to solve.
  • #1
Marin
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Hello everybody!

I have a general question concerning DEs :0

Can one use the symmetry of the equation to somehow get the solution faster?
What does such symmetry tell us?
e.g.:
[tex]\dot x=y[/tex]
[tex]\dot y=x[/tex]

is the symmetrical system to the second order DE

[tex]\ddot x-x=0[/tex]

Now we can easily see the solutions (whether e^t or e^(-t)) actually have the same properties as functions. They are even one and the same function, rotated over the y-axis!

So, is the symmetry really providing help or this is just a coincidence?
 
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  • #2
Marin said:
… So, is the symmetry really providing help or this is just a coincidence?

Hello Marin! :smile:

Well … if x' = y2

y' = x2

then x'' = 2y y' = 2x2 √x'

so that's a symmetry which is no help at all! :cry:

(I suspect there's a condition that makes it helpful … perhaps something like the Jacobian being unitary … but I'll let someone else answer that! :wink:)
 
  • #3
tiny-tim said:
Hello Marin! :smile:

Well … if x' = y2

y' = x2

then x'' = 2y y' = 2x2 √x'

so that's a symmetry which is no help at all! :cry:

(I suspect there's a condition that makes it helpful … perhaps something like the Jacobian being unitary … but I'll let someone else answer that! :wink:)

Sorry,tiny-tim, couldn't quite get it :(

What's the purpose of "then x'' = 2yy' = 2x2 √x'"

When I look at the system itself, what I see is that what's true for y should be true for x which to my understanding implies that the two functions are somehow similar to one another..

And the big question is, if so, then HOW?

**maybe my question above should be: Does the symmetry of a system of simultaneous DEs provide us somehow to find the solution faster?
 
  • #4
Marin said:
When I look at the system itself, what I see is that what's true for y should be true for x which to my understanding implies that the two functions are somehow similar to one another.

oh I see!

Then, yes, both x and y are solutions to the same equation, so they will be different combinations of the same basic solutions. :smile:

(But I don't see how that would generally help.)
 
  • #5
well, if we could find one solution, e.g.:

dy/dx=x^2 => y=1/3 x^3 +c

it is true then that x=1/3y^3 +c

but if x and y are basically the same functions, do we have?:

1/3 x^3=1/3y^3 +k /.3
x^3=y^3 +c

which I think is the solution to the DE, from which the system has been derived, cuz:

the system was:

dx/dt=y^2
dy/dt=x^2

now dividing the second equation by the first one (to eliminate dt):

dy/dx=x^2/y^2 - which is same with the result above.

Was it just a coincidence or is there some symmetry in it?

EDIT: Sorry, I didn't pay attention I used different variables ( first x and then t)
 
  • #6
Hi Marin! :smile:
Marin said:
dx/dt=y^2
dy/dt=x^2

now dividing the second equation by the first one (to eliminate dt):

dy/dx=x^2/y^2 - which is same with the result above.

Was it just a coincidence or is there some symmetry in it?

Yes, I didn't think of that. :redface:

So long as the right-hand side is a function of only one variable,

we can always divide one equation by the other (as you did):

if dx/dt = f'(y), dy/dt = f'(x), say

then f'(y)dy = f'(x)dx,

so f(y) = f(x) + constant. :smile:

You're right … the symmetry does help! :smile:
 
  • #7
And what about the other cases?

consider the system:

[tex]\dot x=x+y^2-2t[/tex]
[tex]\dot y=x^2+y-2t[/tex]

to be honest, I don't have an idea how to solve it analytically :(

But it's symmetrical... You were talking about the Jacobian hmmm could it be an ansatz here ?
 
  • #8
Marin said:
And what about the other cases?

consider the system:

[tex]\dot x=x+y^2-2t[/tex]
[tex]\dot y=x^2+y-2t[/tex]

to be honest, I don't have an idea how to solve it analytically :(

But it's symmetrical... You were talking about the Jacobian hmmm could it be an ansatz here ?

Sorry, I've no idea.

Just guessing about the Jacobian … someone else wil have to answer that. :redface:
 
  • #9
Does anybody know something about it?
 
  • #10
Marin said:
consider the system:

[tex]\dot x=x+y^2-2t[/tex]
[tex]\dot y=x^2+y-2t[/tex]

Hm... look like a challenging problem. Never seen before. Is there any application for this system?

Look like you all been thinking of reflection symmetry [tex] x \leftrightarrow y[/tex] before. May be we should be looking at other transformation such that system remain invariant. Is Lie symmetry is of any used here ? I don't know.

I will monitor this thread. Hopefully somebody could answered it.
 
  • #11
Well, these systems have no physical meaning (at least are not meant to have here). I am interested in the problem from a pure mathematical point of view.

Look like you all been thinking of reflection symmetry [tex] x \leftrightarrow y [/tex] before
- absolutely true - I consider it the most obvious one - if we could find something interesting about it, maybe we could then ask for partial symmetries or negative symmetry, etc.

I know many DEs are not analytically solvable, and many others take a lot of time to find a solution. That's why I'm asking about these symmetrical systems. I think there must be something 'invisible' to us, but hidden in the system.


I would be glad to see every comment or idea - more or less probable :)

best regards, Marin
 

1. What is symmetry in differential equations?

Symmetry in differential equations refers to a property in which the equations remain unchanged when certain transformations are applied. In other words, the equations have a predictable behavior when certain changes are made.

2. What are the benefits of studying symmetry in differential equations?

Studying symmetry in differential equations can help us understand the underlying structure and patterns in the equations. It can also lead to simplification of complex equations and provide insight into the behavior of solutions.

3. How does symmetry affect the solutions of differential equations?

Symmetry can greatly impact the solutions of differential equations. It can lead to the existence of special solutions, such as equilibrium points or periodic solutions. It can also help in finding exact solutions or reducing the dimension of the problem.

4. What are the consequences of neglecting symmetry in differential equations?

Neglecting symmetry in differential equations can lead to incorrect or incomplete solutions. It can also make the problem more difficult to solve and may result in missing important features of the solutions.

5. Can symmetry be applied to all types of differential equations?

Yes, symmetry can be applied to all types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. However, the methods used to analyze symmetry may differ depending on the type of equation.

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