開発 is clearly different from the other two. 開発 means creation, engineering, etc., and 開発する is transitive. コンピュータを開発する means to create/build/engineer computers. As you pointed out, 開発(する) can be used when something new is created.
発展 and 発達 share the similar meaning to a certain extent. コンピュータの発展 and コンピュータの発達 both refer to the advancement of the computer technology in general. Both of these are intransitive when followed by する (i.e. コンピュータが発達する, not *コンピュータを発達する)
The difference of 発展 and 発達 are:
- 発達 is more like sophistication, reaching maturity, increasing complexity, etc., while 発展 is closer to prosperity. For example, 都市の発達 tend to refer to a maturation of infrastructure, economy, or the system of politics, while 都市の発展 tend to refer to an expansion of the territory, increase of the population, etc.
- The development of a body/organ/tumor/etc is almost always 発達. 「子どもの成長と発達」「精神の発達」「イルカの脳は発達している」
- Additionally, 発展 sometimes means "to change", "to evolve." 「事件に発展する (develop into an incident)」「発展問題 (non-straightforward/advanced problem in a textbook)」