How I solve this system?

atomqwerty
How can I solve the differential equations system

dx/dt = y^2 - x^2

dy/dt = -2xy

?

Sorry about not using LaTex, I know it looks better.

Thanks!

Homework Helper
x' = y^2 - x^2
y' = -2xy

Differentiate the first equation

x''= 2yy' - 2x

Use the second equation to eliminate y'
Then use the first equation again to eliminate y

atomqwerty
So I obtain

x'' +4x'x +4x^3 + 2x == 0

and now?

thanks

JJacquelin
Solving in polar coordinates is easier. The main steps only are shown in the attached document.

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atomqwerty
perfect! I got it! Thank you very much :D

Homework Helper
So I obtain

x'' +4x'x +4x^3 + 2x == 0

and now?

thanks

The standard method to solve that type of equation is

Let x' = p
Then x'' = dp/dt = dp/dx dx/dt = p dp/dx
So you get

p dp/dx + 4px + 4x^3 + 2x = 0

Integrating with respect to x gives

p^2/2 + I can't see how to integrate the 4px term + x^4 + x^2 = C

But I like the polar coordinates method better.

Now we know the answer, my method seems to be heading in right direction, but that's no use unless we can see how to finish it.

Homework Helper
Gold Member
In Alephzero's method i think there is a mistake, the 2x term should be 2xx' but this doesn't seem to make things easier.

JJacquelin
In Alephzero's method i think there is a mistake, the 2x term should be 2xx' but this doesn't seem to make things easier.
I agree :
x'' + 6 x x' +4 x^3 = 0
2 y y'' -3 (y')^2 +4 y^4 = 0
OK. far to be easier, but possible, even without knowing the solution.

JJacquelin
Remark : the obvious particular solution [ x=1/(t+c) ; y=0 ] is included in the set of solutions in the particular case of b=c*a and a=> infinity.