Question

y^{\prime \prime}+3 y^{\prime}+4 y=2 \cos 2 t

Differential equations
ANSWERED
asked 2021-05-30

\(\displaystyle{y}^{''}+{3}{y}^{''}+{4}{y}={2}{\cos{{2}}}{t}\)

Answers (1)

2021-05-31

\(\displaystyle{y}{''}+{3}{y}'+{4}{y}={2}{\cos{{2}}}{t}\)
\(\displaystyle{y}{''}=-{3}{y}'-{4}{y}+{2}{\cos{{2}}}{t}\)
We use the substitution
\(y'=v\)
\(\displaystyle\Rightarrow{y}{''}={v}'\)
We substitute y' and y'' in equation
\(\displaystyle{y}{''}=-{3}{y}'-{4}{y}+{2}{\cos{{2}}}{t}\)
we get
\(\displaystyle{v}'=-{3}{v}-{4}{y}+{2}{\cos{{2}}}{t}\)
Therefore, we get system of first-order equations
\(y'=v\)
\(v'=-3v-4y+2\cos 2t\)

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