Second-order linear 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 differential equation $y'' + 4y' + 29y = 0$ has complex-conjugate characteristic roots $\lambda = \alpha \pm i\omega$ (with $\omega > 0$), so the general solution is $y = e^{\alpha x}(c_1\cos(\omega x) + c_2\sin(\omega x))$. Find (i) $\alpha$ and (ii) $\omega$.

2. Warm-up

   The differential equation $y'' - 6y' + 13y = 0$ has complex-conjugate characteristic roots $\lambda = \alpha \pm i\omega$ (with $\omega > 0$), so the general solution is $y = e^{\alpha x}(c_1\cos(\omega x) + c_2\sin(\omega x))$. Find (i) $\alpha$ and (ii) $\omega$.

3. Warm-up

   The differential equation $y'' + 25y = 0$ has complex-conjugate characteristic roots $\lambda = \alpha \pm i\omega$ (with $\omega > 0$), so the general solution is $y = e^{\alpha x}(c_1\cos(\omega x) + c_2\sin(\omega x))$. Find (i) $\alpha$ and (ii) $\omega$.

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