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.

Since $x=0$ is a (regular) singular point of your ODE, if you want a power series solution about $x=0$, you will need to appeal to the Method of Frobenius to construct it.
This is a lengthy process, but not overly hard if you work through the tedium.
Here are some intermediate steps to help you check your work along the way:


*

*The indicial equation is $r(r-1)+{2\over 3}r=0$ so the roots are $r_1=1/3$ and $r_2=0$.

*Since $r_1-r_2\not\in\mathbb{Z}$, the situation is easier than it would have been otherwise. Two linearly independent solutions are
\begin{align}
y_1&=|x|^{r_1}\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty a_nx^{n+1/3},\\
y_2&=|x|^{r_2}\sum_{n=0}^\infty b_nx^n=\sum_{n=0}^\infty b_nx^n.
\end{align}

*To determine the coefficients for each, plug each series back into the ODE and group the coefficients by like powers to obtain a recursion formula for the $n$th coefficient. For a detailed example, see this particular example from the link above.
Hope that helps.
