Heaviside operational calculus

Remarks

In modern terms, $\varphi(p) H$ is the function whose Laplace transform is $\varphi(s)/s$, and rule 2 says that $\mathcal{L}(t^n/n!) = 1/s^{n+1}$. For rational $\varphi$ that is bounded at infinity, the series in rule 4 converges for all $t$ and the method is correct. Heaviside also applied the rules to non-rational expressions such as $\sqrt{p}$, where they need a separate justification.

Example: $x' - a x = a H$, with $x = 0$ for $t < 0$. Rule 3 gives $(p - a) x = a H$, so $x = \frac{a}{p - a} H = \frac{a/p}{1 - a/p} H = \sum_{n \ge 1} a^n p^{-n} H = \sum_{n \ge 1} \frac{a^n t^n}{n!} = e^{at} - 1$ for $t \ge 0$. Check: $x' - ax = a e^{at} - a e^{at} + a = a$ and $x(0) = 0$.

Non-example: the rules do not apply to an equation with variable coefficients such as $x' - t x = H$. There $p$ does not commute with multiplication by $t$, so the equation cannot be solved for $x$ as $\varphi(p) H$.