Question

# Use the substitution v=y' to write each second-order equation as a system of two first-order differential equations (planar system). 4y''+4y'+y=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). $$4y''+4y'+y=0$$

2021-05-13
Let our equation be:
$$\displaystyle{4}{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}+{4}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{y}={0}$$
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:
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}$$
$$\displaystyle{4}{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}+{4}{v}+{y}={0}$$
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}$$
$$\displaystyle{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}=-{\frac{{{1}}}{{{4}}}}{y}-{v}$$
We got system of first order differential equation which solution is equivalent to solution our second order differential equation.
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}$$
$$\displaystyle{F}{\left({x},{v},{y}\right)}=-{\frac{{{1}}}{{{4}}}}{y}-{v}$$