I want to solve this system
\begin{align\*}\tag{\*}
x'(s)=x^2(s)+y(s),
y'(s)=x(s)y(s)
\end{align\*}
with initial conditions $$x(0)=t, y(0)=t,$$
where $t\not=0.$


With the help of Maple, the solution is 


$$ x \left( s \right) ={\frac {2\,st+2\,t}{-{s}^{2}t-2\,st+2}}, ~y \left( s \right) =2\,{\frac {t}{-{s}^{2}t-2\,st+2}}. $$


But I want to know the details of the whole process of solution. 


I have tried to convert this system into a single ODE $$x''(s)=3x(s)x'(s)-x^3(s). \tag{\*\*}$$
But this ODE is semilinear, which is beyond my capability to solve. Then I tried to deduced the order by transformation $$p=x'(s),~~ p\frac{dp}{dx}=x''(s),$$ which simplified $(\*\*)$ to 
$$\frac{dp}{dx}=3x-\frac{x^3}{p},\tag{\*\*\*}$$
which is also difficult for me to solve.


Can anyone help me to solve $(\*)$ , $(\*\*)$ or $(\*\*\*)$ ? 



how to solve this system of nonlinear differential equations
Divide the two equations. You get
$${dx\over dy}={x\over y}+{1\over x}.$$
Multiply by $x$ and set $x^2=u$. You get
$${1\over 2}{du\over dy}={u\over y}+1,$$
a linear equation.


