I've been doing some restoration on an analog computer. While it seems bizarre now to solve a problem with analog circuits instead of a digital computer, analog computers were a reasonable approach until about the 1970s.

The basic idea of an analog computer is you use op amps to sum voltages. Integration is easy, by adding a capacitor. By hooking up op amps, capacitors, and resistors (usually with a plugboard), you can quickly and easily solve differential equations. (Surprisingly, multiplication by a non-constant value is hard with an analog computer.)

The nice thing about analog computers is they gave the answer almost instantaneously. You could turn the dials and experiment with different parameters interactively. A digital computer in 1970, on the other hand, might take minutes to solve the same equation. This also made analog computers useful for real-time applications such as flight simulators.

A big problem with analog computers is that the accuracy is limited by your components. If you want 0.1% accuracy, you need expensive 0.1% resistors and capacitors. With a digital computer, you can get as many bits of accuracy as you want using cheap components. Basically, Moore's law killed off analog computers.