# Solve this system of DE's for the First Integral

1. Oct 25, 2013

### 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!!

2. Oct 25, 2013

### tiny-tim

Hi LBJking123!

Hint: suppose it was

dx/dt = y+x2+y2
dy/dt = x+2xy​

what would you do?

3. Oct 25, 2013

### LBJking123

I tried solving by separation of variables, but I cant 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. Oct 26, 2013

### epenguin

As someone said, you need to pick some different variables. Can you do that for tiny-tim's example?

5. Oct 26, 2013

### tiny-tim

Yes, separation of variables won't work.

Hint: suppose it was

dx/dt = y
dy/dt = x ?​

6. Oct 27, 2013

### 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 wont work for the original DE's though...

7. Oct 28, 2013

### tiny-tim

(just got up :zzz:)
no, all that gives you is x2 - y2 = constant …

how does that help?

try that example again

8. Oct 28, 2013

### 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.

9. Oct 28, 2013

### tiny-tim

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 )

10. Oct 28, 2013

### 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.

11. Oct 28, 2013

### tiny-tim

think laterally!

(but don't give away the answer)