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)

Second order linear equations
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.
$$x_1'=x_2$$
$$x_2'=kx_1-\gamma x_2+F(t)$$

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
$$x_1''=-kx-\gamma x_2+F(t)$$