How can I find two independent solution for this ODE? Please help me find two independent solutions for $$3x(x+1)y''+2(x+1)y'+4y=0$$ Thanks from a  new beginner into ODE's.

Note that both $x=0, x=1$ are regular singular points( check them!). As @Edgar assumed, let $y=\sum_1^{\infty}a_nx^n$ and by writing the equation like $$x^2y''+\frac{2}{3}xy'+\frac{4x}{3(x+1)}y=0$$ we get: $$p(x)=\frac{2}{3},\; q(x)=\frac{4x}{3(x+1)}$$ and then $p(0)=\frac{2}{3},\; q(0)=0$. Now I suggest you to set the characteristic equation (see http://www.math.ualberta.ca/~xinweiyu/334.1.10f/DE_series_sol.pdf) $$s(s-1)+sp(0)+q(0)=0$$ You get $s_1=0,s_2=\frac{1}{3}$. Can you do the rest from here?
