I'm asked to convert the following ODE into a system of order $1$:
$$u''(t) = f(t,u(t),u'(t))$$
and I came up with:
$$\begin{align*}u_0'(t) &= u_1(t)\\u_1'(t) &= f(t,u_0(t),u_1(t))\end{align*}$$
The next task is to show that the Jacobian of the RHS of this system has only real eigenvalues if specific conditions are satisfied, but I'm confused, since the right hand side has no square-matrix as jacobian:
$$\begin{pmatrix}u_0'\\u_1'\end{pmatrix} = F(t,u_0(t), u_1(t))$$
$\implies JF(t,u_0(t),u_1(t)) \in \mathbb{R}^{2 \times 3}$ and there is no corresponding eigenvalue theory. Is my transformation wrong or what am I missing?
As a hint, consider the map $F: \Bbb{R}^3 \to \Bbb{R}^3$ defined by \begin{align} F(x_1,x_2,x_3) = \begin{pmatrix} 1 \\ x_3 \\ f(x_1,x_2,x_3) \end{pmatrix} \end{align}
What you have done so far is to consider $u$ and $u'$ as "separate variables". What the above map does is to consider the time $t$ in $f(t,u(t),u'(t))$ also as an additional variable.