Let our equation be:

\(\displaystyle{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}+{2}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}-{3}{y}={0}\)

Our task is to write our second order differential equation as a system of two first order differential equations. Let's introduce a substitution

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}\) and we get:

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}\)

\(\displaystyle{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}+{2}{v}-{3}{y}={0}\)

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}\)

\(\displaystyle{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}={3}{y}-{2}{v}\)

We got system of first order differential equations which solution is equivalent to solution our second order differential equation.

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}\)

\(F(x,v,y)=3y-2y\)