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
ANSWERED
asked 2021-05-12

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

Answers (1)

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.
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={v}\)
\(\displaystyle{F}{\left({x},{v},{y}\right)}=-{\frac{{{1}}}{{{4}}}}{y}-{v}\)
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