System of differential equation with parameters

Question

I have the system $$\begin{cases}\frac{dx}{dt} = \frac{1}{y}\\ \frac{dy}{dt} = 2xy \end{cases}$$ with $x(0) = 0, y(0) = 1$. Work I've done: I've obtained the equation for $y$ in terms of $x$, which is $$y(x) = \frac{1}{1-x^2}$$ but I cannot figure out how to express $x, y$ in terms of $t$. Any tips or hints on how to proceed would be greatly appreciated!

Answer 1

$$x'=\dfrac 1y$$ You can also differentiate and substitute: $$x''=-\dfrac {y'}{y^2}=-\dfrac 1y \dfrac {y'}{y}=-2x {x'}=-(x^2)'$$ $$x'+x^2=C$$ Since $x'(0)=1$: $$x'+x^2=1$$ This is separable. $$\int \dfrac {dx}{1-x^2}=\int dt$$

Answer 2

Hint: If you take

$$\begin{cases}x(t)&=&\tan(t)\\y(t)&=&\cos^2(t)\end{cases},$$

the two differential equations are verified with the good initial conditions.

What remains to do is to invoke the unicity of solutions of such a system.