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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 $C=0$?

  • 0
    What do we assume on $n$ and $t$ in the equation?2011-09-05
  • 0
    Good question. $t$ is in the domain of $u$ where $u$ is differentiable, so $t \in (0,\infty)$. $n$ is some integer.2011-09-05

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