Power Series Solution to a 2nd order LDE

alain

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Aug 17, 2015
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I am looking for a complete solution of this differential equation: ty"+3y'-3y=0 which appears as problem no 4, section 2.8 of' 'Differential Equations and Their Applications' Fourth edition by Martin Braun (Springer).
I do find one solution y1=Sum (from 0 to infinity) 3^n t^n/(n!(n+2)!), but cannot get to the 2nd solution as shown in the problem description, which appears as y2(t)=y1(t)*ln(t)+1/t^2-1/t+1/4+11*t/36+31*t^2/576+...
I came to y2(t) = -9*y1(t)*ln(t)+1/t^2-3/t+Sum (from 3 to infinity) [3^n*t^(n-2)*(Hn+Hn-2-H2)/(n!(n-2)!)], where Hn is the harmonic function, which is pretty far from the one in the book...Thanks.
 
I am looking for a complete solution of this differential equation: ty"+3y'-3y=0 which appears as problem no 4, section 2.8 of' 'Differential Equations and Their Applications' Fourth edition by Martin Braun (Springer).
I do find one solution y1=Sum (from 0 to infinity) 3^n t^n/(n!(n+2)!), but cannot get to the 2nd solution as shown in the problem description, which appears as y2(t)=y1(t)*ln(t)+1/t^2-1/t+1/4+11*t/36+31*t^2/576+...
I came to y2(t) = -9*y1(t)*ln(t)+1/t^2-3/t+Sum (from 3 to infinity) [3^n*t^(n-2)*(Hn+Hn-2-H2)/(n!(n-2)!)], where Hn is the harmonic function, which is pretty far from the one in the book...Thanks.


What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting
 
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