Question 113
g is a twice differentiable function over the positive reals such that
$$
\begin{aligned}
g(x)+2 x^{3} g^{\prime}(x)+x^{4} g^{\prime \prime}(x) & =0 \quad \text{ for all positive reals } x . \\
\lim _{x \rightarrow \infty} x g(x) & =1
\end{aligned}
$$
Find the real number $\alpha\gt 1$ such that $g(\alpha)=1 / 2$.
$$
\begin{aligned}
g(x)+2 x^{3} g^{\prime}(x)+x^{4} g^{\prime \prime}(x) & =0 \quad \text{ for all positive reals } x . \\
\lim _{x \rightarrow \infty} x g(x) & =1
\end{aligned}
$$
Find the real number $\alpha\gt 1$ such that $g(\alpha)=1 / 2$.