Find the basic solutions at $\displaystyle x_{0} = 1

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$\displaystyle x^2\varphi''+x\varphi'+\lambda\varphi=0$

$\displaystyle x^m\left[m(m-1)+m+\lambda\right]=0\Rightarrow x^m\left[m^2+\lambda\right]=0$

$\displaystyle m=\pm i\sqrt{\lambda}$

$\displaystyle \varphi(x)=C_1\cos\left(\sqrt{\lambda}\ln(x)\right )+C_2\sin\left(\sqrt{\lambda}\ln(x)\right)$

How do I find the basic solutions around $\displaystyle x_{0} = 1\text{?}$