1) d4y/ dt4 –2d3y/ dt3 –7 d2y/ dt2 + 20 dy/dt –12 y = 0
Rewrite as: yiv –
2y”’ - 7y” + 20y’ – 12 y = 0
This can be factored (i.e. guess or use synthetic division) to get:
(m-1) (m- 2)2 (m-3)
Yielding the roots: m = 1, m = 2 (twice) and m = 3
So, the general solution is:
y= c1 exp (x) + (c2 + c3) exp(2x) + c4 exp (-3x)
2) y”’ + 3y” – 4y = 0
The auxiliary equation here is:
m3 + 3m2
– 4 = 0
Use synthetic division, careful inspection (or guessing) to obtain
the factors:
Yielding the roots: m = 1, m = 2 (twice), so the general
solution is:
y = c1 exp (x) + c2 exp(-2x) + c3(x exp (-2x))
(Note that the absence of a 1st derivative term, i.e. dy/dt or y’, means one of the roots will be in terms of x exp (r2x) where r2 is the 2nd of the twinned roots)
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