How Do You Find the First Integral of This System of Differential Equations?

In summary, LBJking123 is trying to solve a system of DE's to determine the First Integral, but he is having trouble. He has tried different variables, but hasn't found a solution yet. He is hoping somebody else can help him.
  • #1
LBJking123
13
0
I am trying to solve this system DE's to determine the systems First Integral.

dx/dt = y+x2-y2
dy/dt = -x-2xy

I am pretty sure I need to pick some different variables to use to make the equation easier to solve, but I can't get anything to work. I thought about letting a variable be x2y, but that doesn't help much. If anyone can help me that would be much appreciated!
 
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  • #2
Hi LBJking123! :smile:

Hint: suppose it was

dx/dt = y+x2+y2
dy/dt = x+2xy​

what would you do? :wink:
 
  • #3
I tried solving by separation of variables, but I can't figure out how to get all of the x's to one side and y's to the other. I think I am totally missing something obvious...
 
  • #4
LBJking123 said:
I tried solving by separation of variables, but I can't figure out how to get all of the x's to one side and y's to the other. I think I am totally missing something obvious...

As someone said, you need to pick some different variables. Can you do that for tiny-tim's example?
 
  • #5
LBJking123 said:
I tried solving by separation of variables, but I can't figure out how to get all of the x's to one side and y's to the other.

Yes, separation of variables won't work.

Hint: suppose it was

dx/dt = y
dy/dt = x ?​
 
  • #6
That case you could divide the two equations, and then get xdx=ydy. Then I would integrate both sides to get the answer. That technique won't work for the original DE's though...
 
  • #7
(just got up :zzz:)
LBJking123 said:
That case you could divide the two equations, and then get xdx=ydy. Then I would integrate both sides to get the answer.

no, all that gives you is x2 - y2 = constant …

how does that help? :redface:​

try that example again :smile:
 
  • #8
I am not QUITE sure where tiny-tim is trying to lead you. LBJking123, but I, at least, felt that the variable change u=x+y and v=x-y simplifies the equations in a manner that may be amenable for further simplifications.
 
  • #9
arildno said:
I am not QUITE sure where tiny-tim is trying to lead you. LBJking123, but I, at least, felt that the variable change u=x+y and v=x-y simplifies the equations in a manner that may be amenable for further simplifications.

hi arildno! :smile:

yes, that would be the way to solve my easy example (but i was hoping LBJking123 would see it on on his own :redface:)
 
  • #10
Well, I spotted some further troubles on the way (I was hoping a g(u/v) substitution would turn up, but it doesn't seem to be THAT simple..)
So, I have been following this thread for a while, and am hoping to see some real cleverness on your part in the end that I have missed.
:smile:
 
  • #11
think laterally! :wink:

(but don't give away the answer)
 

Related to How Do You Find the First Integral of This System of Differential Equations?

1. What is a system of DE's?

A system of DE's (differential equations) is a set of equations that describe the relationship between a dependent variable and one or more independent variables, along with their respective derivatives. In other words, it is a mathematical model used to represent the behavior of a system over time.

2. What is a First Integral?

A First Integral is a constant of motion that remains constant throughout the solution of a system of DE's. It is often used to simplify the equations and find a solution without having to solve the DE's directly.

3. How do you solve a system of DE's for the First Integral?

To solve a system of DE's for the First Integral, you can use the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding a solution that satisfies the DE's and then using the initial conditions to determine the First Integral.

4. Why is the First Integral important in solving a system of DE's?

The First Integral is important because it allows us to reduce a system of DE's into a simpler form, making it easier to find a solution. It also helps to characterize the behavior of the system and determine any constants or relationships between variables.

5. Can a system of DE's have more than one First Integral?

No, a system of DE's can only have one First Integral. This is because the First Integral is a constant of motion and therefore should remain constant for all solutions to the system. If there were multiple First Integrals, they would not remain constant and would contradict the nature of a First Integral.

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