We tend to think of computers as having been developed starting in the years after the second world war, but in the late 19th and early 20th centuries the focus was on analog computers, which were seen as being the way of the future.

Wikipedia gives examples of machines capable of integrating differential equations starting in 1836, and leading to:

The first description of a device which could integrate differential equations of any order was published in 1876 by James Thomson, who was born in Belfast in 1822, but lived in Scotland from the age of 10.^[5] Though Thomson called his device an “integrating machine”, it is his description of the device, together with the additional publication in 1876 of two further descriptions by his younger brother, Lord Kelvin, which represents the invention of the differential analyser.^[6]

One of the earliest practical uses of Thomson’s concepts was a tide-predicting machine built by Kelvin starting in 1872–3. On Lord Kelvin’s advice, Thomson’s integrating machine was later incorporated into a fire-control system for naval gunnery being developed by Arthur Pollen, resulting in an electrically driven, mechanical analogue computer, which was completed by about 1912

https://en.wikipedia.org/wiki/Differential_analyser

The Water Integrator was an early analog computer built in the Soviet Union 1936 by Vladimir Sergeevich Lukyanov. This was of particular interest to me, because of an early application:

Lukyanov was one of the engineers working on the construction of the Troitsk-Orsk and Kartaly-Magnitnaya railways in the late 1920s. To ensure the quality and durability of reinforced concrete structures, the engineers poured concrete only in the summer. Despite this, cracks still appeared in the concrete when temperatures dropped below zero in winter. Lukyanov suggested that this can be avoided if a careful analysis of the temperature changes in the concrete mass is made, depending on the composition of the concrete, the cement used, the technology of the work, and the external conditions. Lukyanov began studying temperature conditions in concrete masonry, but the existing calculation methods could not give a quick and accurate solution to the complex differential equations that described the temperature regime.

n search of a new approach to solving the problem, Lukyanov discovered that water flow is in many respects similar in its laws to the distribution of heat. He concluded that by building a computer where the main component was water, Lukyanov could visualize the invisible thermal process. In 1936, Lukyanov built the first model of his “water integrator” at the Institute of Way and Construction (now Central Research Institute of Transport Construction, or TsNIIS). At that time it was the only computer that could solve partial differential equations.

https://www.amusingplanet.com/2019/12/vladimir-lukyanovs-water-computer.html

For more details and photographs see: Vladimir Lukyanov’s Water Computer.