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

[LBJking123]

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!

 

[tiny-tim]

Hi LBJking123!

Hint: suppose it was

dx/dt = y+x²+y²
dy/dt = x+2xy​

what would you do?

 

[LBJking123]

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...

 

[epenguin]

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?

 

[tiny-tim]

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 ?​

 

[LBJking123]

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...

 

[tiny-tim]

(just got up :zzz:)

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? ​

try that example again

 

[arildno]

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.

 

[tiny-tim]

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!

yes, that would be the way to solve my easy example (but i was hoping LBJking123 would see it on on his own )

 

[arildno]

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.

 

[tiny-tim]

think laterally!

(but don't give away the answer)