Section 7.1 – Introduction [to Systems of 1st Order Equations]

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Transform the given initial value problem into an initial value problem for three first-order equations, and write it in matrix form. $u'''+p(t)u''+q(t)u'+r(t)u=g(t), \quad u(0)=u_0, \quad u'(0)=u_1, \quad u''(0)=u_2$

$$\overrightarrow{X'}=\begin{pmatrix}0&1&0\\0&0&1\\-r(t)&-q(t)&-p(t)\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}+\begin{pmatrix}0\\0\\g(t)\end{pmatrix}, \quad$$ $$\overrightarrow{X}(0)=\begin{pmatrix}u_0\\u_1\\u_2\end{pmatrix}$$

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