Systems of differential equations

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   The second-order equation $y'' + 2y' + 5y = 0$ is converted to a first-order system by setting $x_1 = y$, $x_2 = y'$, giving $x_1' = x_2$, $x_2' = -5x_1 - 2x_2$, i.e. $\mathbf{x}' = A\mathbf{x}$ with $A = \begin{pmatrix} 0 & 1 \\ -5 & -2 \end{pmatrix}$. Compute $\det(A)$.

2. Warm-up

   The second-order equation $y'' + 5y' + 2y = 0$ is converted to a first-order system by setting $x_1 = y$, $x_2 = y'$, giving $x_1' = x_2$, $x_2' = -2x_1 - 5x_2$, i.e. $\mathbf{x}' = A\mathbf{x}$ with $A = \begin{pmatrix} 0 & 1 \\ -2 & -5 \end{pmatrix}$. Compute $\det(A)$.

3. Warm-up

   The second-order equation $y'' + 3y' + 5y = 0$ is converted to a first-order system by setting $x_1 = y$, $x_2 = y'$, giving $x_1' = x_2$, $x_2' = -5x_1 - 3x_2$, i.e. $\mathbf{x}' = A\mathbf{x}$ with $A = \begin{pmatrix} 0 & 1 \\ -5 & -3 \end{pmatrix}$. Compute $\det(A)$.

Problems shown under their original terms: Generated (template, solver-computed).