Question

[Differential Equations]: Intro to Systems

Please see the comments for the question. I mainly need help with just conceptualizing this, I really don't know where to begin.

Answer 1

Write the initial value problem \[y'' - 3y' + 2y = e ^{3t}; y _{o} = 4; y' _{o} = 5\]as a first-order system: \[\left(\begin{matrix}x _{1} \\ x _{2}\end{matrix}\right)' = \left(\begin{matrix}a _{11} & a _{12} \\ a _{21} & a _{22}\end{matrix}\right)\left(\begin{matrix}x _{1} \\ x _{2}\end{matrix}\right) + \left(\begin{matrix}g _{1}(t) \\ g _{2}(t)\end{matrix}\right); \left(\begin{matrix}x _{1}(0) \\ x _{2}(0)\end{matrix}\right) = \left(\begin{matrix}b _{1} \\ b _{2}\end{matrix}\right)\].

Answer 2

Suppose you set \(x_1=y\), then \({x_1}'=y'\). Now set \(x_2=y'={x_1}'\), then \({x_2}'=y''\).

Substituting into the ODE, you have
\[\begin{cases}{x_2}'-3x_2+2x_1=e^{3t}\\{x_1}'=x_2\end{cases}\]
Written as a matrix equation, you have
\[\begin{pmatrix}x_1\\x_2\end{pmatrix}'=\begin{pmatrix}0&1\\-2&3\end{pmatrix}\begin{pmatrix}x_1\\x_2\end{pmatrix}+\begin{pmatrix}0\\1\end{pmatrix}e^{3t}\]
with initial conditions \(\begin{pmatrix}x_1(0)\\x_2(0)\end{pmatrix}=\begin{pmatrix}4\\5\end{pmatrix}\).