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
ANSWERED
asked 2021-05-23

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

Expert Answers (1)

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

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