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 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.