Let $u:[0,\infty) \to \mathbb{R}$ be a continuously differentiable function in $t$, and let t^{n-1} u'(t) + \frac{1}{2} t^n u(t) = C for some constant $C$ and positive integer $n$.
Suppose that $\displaystyle\lim_{t\to +\infty} u(t) = 0$ and \displaystyle\lim_{t\to +\infty} u'(t) = 0.
How can one show that then $C=0$?