math derivative 4

s = t / (t - 1)^(1/2)
ds/dt = t(-1/2)(t-1)^(-3/2) + 1(t-1)^(-1/2)
= (t-1 - t/2) (t-1)^(-3/2)
= [(t-2)/2] (t-1)^(-3/2)
d/dt(ds/dt) = (1/2)(t-2) (-3/2) (t-1)^(-5/2) + (1/2)(t-1)^(-3/2)
= (1/2)(-3t/2 + 3 + t-1) (t-1)^(-5/2)
= (1/2) (-t/2 + 2) (t-1)^(-5/2)
= (1/4) (4-t) (t-1)^(-5/2)

2t(t-1)d/dt(ds/dt)
= (t/2) (4-t) (t-1)^(-3/2)

3t(ds/dt) = (3t/2)(t-2) (t-1)^(-3/2)
2t(t - 1)(d/dt(ds/dt)) + 3t(ds / dt)
= (t/2)(4-t) (t-1)^(-3/2) + (3t/2)(t-2) (t-1)^(-3/2)
= (t/2) [ (4-t) + 3(t-2)] (t-1)^(-3/2)
= (t/2) (2t-2) (t-1)^(-3/2)
= t (t-1) (t-1)^(-3/2)
= t / (t-1)^(1/2)
= s

Since s' = (t-2)/[2(t-1)^3/2]
= [(t-1) - 1]/[2(t-1)^3/2]
= 1/[2(t-1)^1/2] - [1/2(t-1)(t-1)^1/2]
= s/2t - s/(2t^2 - 2t)
that is
(2t^2 - 2t)s' = s(t-2).
Differentiate one more time we get
(2t^2 - 2t)s" + (4t - 2)s' = (t-2)s' + s
2t(t-1)s" + 4ts' - 2s' = ts' - 2s' + s
so 2t(t-1)s" + 3ts' = s.

圖片參考：http://i388.photobucket.com/albums/oo325/loyitak1990/Feb09/Crazydiff1.jpg

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