Who was first to solve equations of the form $y' + p(t)y = g(t)$? The method of the integration factor is mildly tricky to students at first, so I imagine there must have been some time spent to come up with that trick. I know that Bernoulli solved the so called Bernoulli's equation $y' + p(t)y = g(t)y^n$ in 1695, so it must have been some time before that.
2026-04-06 16:34:16.1775493256
Who was the first to solve the linear first order ODE?
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Leibniz, in 1694.
Leibniz' method is explained by Victor Katz, and it is the familiar integrating factor method with some slight changes in notation. For example, Leibniz defines the integrating factor $u(t)$ as the solution to the equation ${{\rm d}u\over u}=p\,{\rm d}t$, instead of the today more familiar $u(t)={\rm exp}\bigl(\int p(t)\,{\rm d}t\bigr)$.