I'm trying to find other linear independent solution given $J_0(x)$ is the solution of the Bessel's equation.
Bessel's equation looks like $$x^2y''+xy'+(x^2-n^2)y=0$$ Now it's given $n=0$ $$x^2y''+xy'+x^2y=xy''+y'+xy=0$$ Now I'm thinking to solve it with frobenius method. But that would leave to again $J_0(x)$.
I might write it directly by relation $$Y_n(x)=\frac{J_n(x)\cos(n\pi)-J_{-n}(x)}{\sin(nx)}$$ But How do I find not knowing this?
The solution of $$x^2y''+xy'+x^2y=xy''+y'+xy=0$$ is given by $$y=c_1 J_0(x)+c_2 Y_0(x)$$ You cannot find the $Y_0(x)$ using Frobenius method since they cannot be expanded as polynomial.
Take care that $$Y_\alpha(x) = \frac{J_\alpha(x) \cos (\alpha\pi) - J_{-\alpha}(x)}{\sin (\alpha\pi)}$$ only applies for non-integer values of $\alpha$.
For integer values of $n$, the expansion of $Y_0(x)$ is given as $$Y_0(x)=\frac 1 \pi\sum_{n=0}^\infty (a_n+b_n \log(x))\, x^{2n}$$