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MOTIVATION. Let $f:[0,+\infty)\to \mathbb{R}$. Solutions to $\partial_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero exactly as $e^{-\lambda t}$. This property is preserved if we apply an ...

The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function). $$(x^2y')'-x^2y=\lambda \;y$$ Now for a higher-degree ...

Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is $$ u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y) $$ ...