# Exact solution for a first order autonomous algebraic ordinary differential equation

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.

• Indeed, thank you for noticing I adapted it. – Darkwizie Oct 5 '18 at 12:15
• I hope it does! – Darkwizie Oct 5 '18 at 12:17
• I am mainly interested in which techniques COULD be applied to find a solution, I can try to apply these methods myself but when I google myself I am overwhelmed by overly complicated methods and have no idea which of them could work well. – Darkwizie Oct 5 '18 at 12:24

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.