Solution
= Solution
Expanding the derivative and dividing by $1-x$ gives
$$
y''-\frac1{1-x}y'+\left(c^2+\frac\lambda{1-x}\right)y=0.
$$
The only finite singular point is $\boxed{x=1}$. There $(x-1)p(x)=1$ and $(x-1)^2q(x)=c^2(x-1)^2-\lambda(x-1)$ are analytic, so it is a <regular singular point>. At infinity the coefficient of $y$ tends to $c^2\ne0$, giving an <irregular singular point> after reciprocal substitution; the finite classification is independent of $\lambda$.