June 1st, 2014, 06:47 PM  #1 
Member Joined: Sep 2013 Posts: 43 Thanks: 0  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. 
June 1st, 2014, 07:47 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2588 Math Focus: Mainly analysis and algebra 
$$\frac{d^2h}{dt^2} = \frac{d}{dt} \frac{dh}{dt} = \frac{d}{dt} h^2 = 2h\frac{dh}{dt}$$ 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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