Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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Solution of differential equation related to Normal density

Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ Given $0<\sigma\le 1$. I wish to know whether there ...
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First order differential equation involving inverse function

I am wondering if there is a way to solve a differential equation of the following form: $$\displaystyle \frac{f'(x)}{x} = \frac{1}{f^{-1}(x)} + \frac{1}{k}$$ We can assume that $f(x): [0,T] \to ...