Say I have an ODE $\frac{dy}{dx} + \frac{dz}{dx} + a(y) + b(z) = 0$ Where $a$ is a function of $y$ and $b$ a function of $z$, then how do I solve this ODE for $y$ and $z$?
Can I split it in to two ODEs $\frac{dy}{dx} + a(y) = 0$ and $\frac{dz}{dx} + b(z) = 0$?
Then will the solution just be the sum of those solutions from the above two?
Your procedure will give only a subset of all solutions. You can choose $z$ arbitrarily, and then find $y$ by solving the equation $$ y'+a(y)=-z'-b(z). $$