Find two series solutions of the form $$ x^{1/2}\sum_{n=0}^{\infty }a_nx^n\\ x^{-1/2}\sum_{n=0}^{\infty }b_nx^n $$ of the Bessel equation of order $\alpha=\frac{1}{2}$:
$$ x^2y''+xy'+(x^2-\alpha^2)y=0\\ u=x^{\frac{1}{2}}y\\ u''+\left(1+\frac{1/4-\alpha^2}{x^2}\right)u=0\\ u''+u=0\\ u=\cos x, u=\sin x\\ y=x^{-1/2}\cos x,y=x^{-1/2}\sin x\\ y=x^\alpha\left[1+\sum_{m=1}^{\infty }\frac{(-1)^m}{(\alpha+1)\cdots(\alpha+m)m!}\left(\frac{x}{2}\right)^{2m}\right]$$ $$y_2=x^{-\alpha}\left[1+\sum_{m=1}^{\infty }\frac{(-1)^m}{(\alpha+1)\cdots(\alpha+m)m!}\left(\frac{x}{2}\right)^{2m}\right]$$
But where to from here?