Question

# Use the substitution v=y' to write each second-order equation as a system of two first-order differential equations (planar system). y''+2y'-3y=0

Differential equations

Use the substitution $$v=y'$$ to write each second-order equation as a system of two first-order differential equations (planar system). $$y''+2y'-3y=0$$

2021-05-24

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