(1) Let $C_1$ and $C_2$ be arbitrary constants. The general solution to the homogeneous differential equation $x^2 y''+ 35 x y' + 289 y = 0$ is the function $y(x)=C_1\ y_1(x)+C_2\ y_2(x)=C_1$ $+C_2$.

(2) The unique solution to the initial value problem $$x^2 y''+35 x y' + 289 y =0,\ \ y(1) = 2,\ \ y'(1) = -7.$$ is the function $y(x)=$ for $x \in$ .
For $-\infty$ type -inf and for $\infty$ type inf.