Second and Higher Order Linear Differential Equations

Solutions.
1. Solutions of x 4 = 1 : ±1, ±i. Solutions of x 8 = 1 : ±1, ±i, ±1/ √ 2±i1/ √ 2, ±1/ √ 2∓i1/ √ 2.
2. 2, −1 ± i √ 3.
3. 2, 2(cos 2π/5 ± i sin 2π/5), 2(cos 4π/5 ± i sin 4π/5).
4. Solutions of x 4 = 1 are ±1, ±i. Thus, the general solution of the differential equation is
y = c1e t + c2e −t + c3 cos t + c4 sin t.
5. Solutions of x 3 = 8 are 2, −1±i √ 3. Thus, the general solution of the differential equation
is y = c1e 2t + c2e −t cos √ 3t + c3e −t sin √ 3t.
6. The characteristic equation r 4 − 5r 2 − 36 factors as (r 2 − 9)(r 2 + 4). So, the solutions are
3, −3, 2i, −2i. Thus the general solution is y = c1e 3x + c2e −3x + c3 cos 2x + c4 sin 3x.
7. The characteristic equations corresponds to a polynomial p that can be represented in
Matlab as p=[-18 25 0 -27 16 20]. The command roots(p) gives you the following
values: 1.2971, 0.7664 ± 0.9707i and −0.7205 ± 0.2023i. Thus, the general solutions
is y = c1e 1.2971x + c2e 0.7664x cos 0.9707x + c3e 0.7664x sin 0.9707x + c4e −0.7205x cos 0.2023x +
c5e −0.7205x sin 0.2023x.
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