Izabella Ponce

2022-06-20

We know that every differential equation is equivalent to a first-order system. I am trying to prove or disprove the converse. For example in ${\mathbb{R}}^{2}$, if we have a system $\stackrel{˙}{x}=f\left(x,y\right)$, $\stackrel{˙}{y}=g\left(x,y\right)$. Can we always convert it to one differential equation (for example, only in terms of $x$)? Under what condition, this is possible?

Abigail Palmer

Consider your example. In order to make this a second-order equation in $x$, you want to solve $\stackrel{˙}{x}=f\left(x,y\right)$ for $y$ as a function of $x$ and $\stackrel{˙}{x}$, say $y=a\left(x,\stackrel{˙}{x}\right)$. This may or may not be possible, (and in most cases even if it is possible in principle it can't be done in closed form). Then we get
$\stackrel{¨}{x}={f}_{1}\left(x,y\right)\stackrel{˙}{x}+{f}_{2}\left(x,y\right)\stackrel{˙}{y}={f}_{1}\left(x,a\left(x,\stackrel{˙}{x}\right)\right)\stackrel{˙}{x}+{f}_{2}\left(x,a\left(x,\stackrel{˙}{x}\right)\right)g\left(x,a\left(x,\stackrel{˙}{x}\right)\right)$
where ${f}_{1}$ and ${f}_{2}$ are the partial derivatives of $f$.

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