Solving two coupled ODEs without use of complex numbers

Question

Consider a system of ODEs where $x, y$ are functions to be determined.
$$ x'(t) = 2x(t) y(t) $$
$$ y'(t) = y^2(t) - x^2(t) $$
It is pretty straightforward to solve this system by introducing complex function $z(t) = x(t) + i y(t)$, but I would like to solve this problem without use of complex analysis. I have tried ordinary technique of reducing this to single function but second order differential equation. I get the following for $x(t)$.
$$ 2x''(t) x(t) - 3 (x'(t))^2 + 4x^4(t) = 0 $$
But then, I am not sure on how to solve this.
I believe that the correct solution is the following.
$$ x(t) = frac{x(0)}{(1-ty(0))^2 + (tx(0))^2} $$
$$ y(t) = frac{y(0)-t((x(0))^2-(y(0))^2)}{(1-ty(0))^2 + (tx(0))^2} $$

JJacquelin · Accepted Answer

$$begin{cases}
frac{dx}{dt}=2xy\
frac{dy}{dt}=y^2-x^2
end{cases} $$
$$frac{dy}{dx}=frac{y^2-x^2}{2xy}$$
$$frac{d(y^2)}{dx}=2yfrac{dy}{dx}=frac{y^2-x^2}{x}=frac{y^2}{x}-x$$
Let $u(x)=y^2$
$$frac{du}{dx}-frac{u}{x}=-x$$
This is a first order linear ODE which method of solving is well known.
$$u(x)=c_1:x-x^2$$
$$boxed{y(x)=pmsqrt{c_1:x-x^2}}tag 1$$
$$frac{dx}{dt}=pm 2xsqrt{c_1:x-x^2}$$
$$dt=pmfrac{dx}{2xsqrt{c_1:x-x^2}}$$
$$t=pmint frac{dx}{2xsqrt{c_1:x-x^2}}=pm sqrt{frac{c_1-x}{x}}+c_2$$
$$boxed {x(t)=frac{c_1}{1+(t-c_2)^2}} tag 2$$
Then put Eq.$(2)$ into Eq.$(1)$ in order to find $y(t)$ .

Solving two coupled ODEs without use of complex numbers

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