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Solving two coupled ODEs without use of complex numbers

Mathematics Asked by Daniels Krimans on November 27, 2020

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.

$$ x(t) = frac{x(0)}{(1-ty(0))^2 + (tx(0))^2} $$
$$ y(t) = frac{y(0)-t((x(0))^2-(y(0))^2)}{(1-ty(0))^2 + (tx(0))^2} $$

One 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 $y(t)$ .

Correct answer by JJacquelin on November 27, 2020

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