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June 1st, 2014, 05:47 PM   #1
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Taylor series

I have a differential equation dh/dt=-h^2 and h(0)=h0

exact solution is h(t)=h0/(th0+1)

but if written in taylor expansion is h(0+t)=h0+h'(0)(t)+t^2h"(0)/2 and so on

how to find the second derivative and so on? quite confuse on this.
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June 1st, 2014, 06:47 PM   #2
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$$\frac{d^2h}{dt^2} = \frac{d}{dt} \frac{dh}{dt} = \frac{d}{dt} -h^2 = -2h\frac{dh}{dt}$$
by the chain rule.

You know $\frac{dh}{dt}$ already, so you can substitute that in and proceed as far as you need to.
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