I have the following system of ordinary differential equations: \begin{align*} x'&=\frac{1}{4}x^2+\frac{3}{4}y^2-2x\\ y'&=\frac{1}{12}x^2+\frac{1}{4}y^2-\frac{2}{3}y, \end{align*} with boundary condition $x(0)=y(0)=1$. Note that $\frac{1}{2}x'-\frac{3}{2}y'=y-x$.
After some googling I found that this is an Autonomous Algebraic Ordinary Differential Equation (AODE), and there exists a large body of literature for finding exact solution for these type of ODEs. I was wondering if someone could point me in the right direction for solving this specific AODEs.
You found that $$ x=3y+Ce^{-t}. $$ Inserting into the second equation results in $$ y'=y^2+\frac12Ce^{-t}y-y+\frac1{12}C^2e^{-2t} $$ This is now a Riccati equation where you can use the standard substitution $y=-\frac{u'}{u}$ to obtain a second order linear ODE.