2nd basis function for 2nd order ODE

[X89codered89X]

i have the first solution y_1(t) = t for (1-t)y'' + ty' - y = 0.

I need to get the 2nd linearly independent using Abels theorem.

the integration is messy but i have it set up (sorry no latex);

y_2 = (t) * integral to t ( 1/s^2 * exp( -integral to t (s(s+1) ds) ) ds.

Could anyone show me how to do this integration step by step?

Thanks in advance!

[X89codered89X]

found it, integral was a telescoping series from parts. the solution is \exp(-t)

[JJacquelin]

Obviously, the second solution is exp(t)