Write u''+0.5u'+2u=3 sin t as a system of first order questions. Write the systemof first order differential equations as a second order differential equation. x_1'=x_2 x_2'=kx_1-gamma x_2+F(t)

Write u''+0.5u'+2u=3 sin t as a system of first order questions. Write the systemof first order differential equations as a second order differential equation. x_1'=x_2 x_2'=kx_1-gamma x_2+F(t)

Question
Write u''+0.5u'+2u=3 sin t as a system of first order questions.
Write the systemof first order differential equations as a second order differential equation.
\(x_1'=x_2\)
\(x_2'=kx_1-\gamma x_2+F(t)\)

Answers (1)

2020-11-24
Given differential equation is ,
\(u''+0.5u'+2u=3\sin t\)
Let, \(u'=v\) then,
\((u')'+0.5u'+2u=3\sin t\)
\(\Rightarrow v'+0.5v+2u=3\sin t\)
\(\Rightarrow v'=3\sin t-2u-0.5v\)
Hence,
\begin{cases}u'=v\\v'=3\sin t-2u-0.5v\end{cases}\)
Given system of first order differential equations.
\(x_1"=x_2\)
\(x_2'=-kx_1-\gamma x_2+F(t)\)
\(\Rightarrow x_1''=-kx-\gamma x_2+F(t)\)
Hence, the required second order differential equation is
\(x_1''=-kx-\gamma x_2+F(t)\)
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