Chapter 12.21, Problem 6P

Solve the differential equations in Problems 5 to 10 by the Frobenius method; observe that you get only one solution. (Note, also, that the two values of $s$ are equal or differ by an integer, and in the latter case the larger $s$ gives the one solution.) Show that the conditions of Fuchs’s theorem are satisfied. Knowing that the second solution is $\ln x$ timesthe solution you have, plus another Frobenius series, find the second solution.

$4x^2(x+1)y^{\prime \prime }-4x^2{y}^{\prime }+(3x+1)y=0$