Laplace's equation "find a solution of an equation inside the spherical layer"

Question

This is a problem from Russian book and I'm not sure what is asked for. This is the translation I came up with. "Find a solution of an equation inside the spherical layer with the following boundaries" $a<r<b$ $$\Delta u=1$$ the boundaries$$u|_{r=a}=0$$ $$u|_{r=b}=0$$ I have done the following $$\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial u}{\partial r})=1$$ and when I integrate $$u=\frac{1}{6}r^2-\frac{C_1}{r}+C_2$$ $$u_{r=a}=\frac{1}{6}a^2-\frac{C_1}{a}+C_2=0$$ $$u_{r=b}=\frac{1}{6}b^2-\frac{C_1}{b}+C_2=0$$ Next I think that I need to find the values of $C_1$ and $C_2$. If I set $$u_{r=a}=u_{r=b}$$ and solve for $C_1$ and $C_2$ $$C_1=\frac{1}{6}ab(a-b)=\frac{1}{6}r^2(r-r)=0$$ $$C_2=\frac{1}{6}a(a-b)-\frac{1}{6}b^2=\frac{1}{6}r(r-r)-\frac{1}{6}r^2=-\frac{1}{6}r^2$$ If I substitute $C_1$ and $C_2$ into the original equation I will get zero. What am I doing wrong here?