BVP where $u$ is finite as $x\rightarrow 0^+$

Question

I am trying to solve the boundary value problem $$x^2u''+2xu'-2u=18x^4 \ \ \ \ 0<x<2,$$ $$u \ \text{is finite, as } x\rightarrow 0^+$$ $$u'-u=0 \ \ \ \ \text{at} \ x=2$$

My attempt:

I have shown that the general solution is $$u(x)=u_H(x)+u_P(x)=C_1x^{-2}+C_2x+x^4.$$ I have also shown that the second condition yields $$C_1+2C_2=32.$$ However, I am unsure how to utilise the first condition. How can $u$ be finite when $C_1x^{-2}\rightarrow\infty$ as $x\rightarrow 0^+$? A hint would be very helpful.

Answer 1

$ \lim_{x \to 0+} u(x)$ finite implies $C_1=0$ and this gives $C_2=16$.