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November 28th, 2017, 09:49 PM  #1 
Newbie Joined: Nov 2017 From: NY Posts: 1 Thanks: 0  Solve with Series
Hi! I've been working at this for hours now and can't get it. I need to find the first 8 coefficients (A0A7) of the power series solution x(t) = SUM[An (t^n) ] of the initial value problem x′′ +tx′ −(e^t)x=0, x(0)=1, x′(0)=−1 Obviously, A0 = 1 and A1 = 1. From that I found A2=1 but I'm really stumped after that. I've tried combining series as much as I can. From all the examples in my textbook, it should condense to one series but I can't get it to do that. PLEASE help thank you 
December 8th, 2017, 07:41 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
Well, what have you tried? Writing, as you did x= A0+ A1t+ A2t^2+ A3t^3+ A4t^4+ A5t^5+ A6t^6+ A7t^7, x'= A1+ 2A2t+ 3A3t^2+ 4A4t^3+ 5A5t^4+ 6A6t^5+ 7A7t^6 and x''= 2A2+ 6A3t+ 12A4t^2+ 20A5t^3+ 30A6t^4+ 42A7t^5. Of course, e^t= 1+ t+ t^2/2+ t^3/6+ t^4/24+ t^5/120+ t^6/720+ t^7/5040 so that e^t y= (1+ t+ t^2/2+ t^3/6+ t^4/24+ t^5/120+ t^6/720+ t^7/5040)(A0+ A1t+ A2t^2+ A3t^3+ A4t^4+ A5t^5+ A6t^6+ A7t^7). Doing that multiplication is the "tedious" part. Notice that would give a highest power of t^14. But you can ignore anything after t^7. 

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