There are several numerical methods for solving IVPs (of ODEs). There are single step or multi-step methods. The methods may be classified as explicit and implicit methods. When applying numerical methods convergence, consistency and stability are among important points of considerations.
Implicit vs. explicit algorithms for solving IVPs, ODEs, and PDEs, is a topic often discussed in the literature. Like other similar topics, some authors may make much of one method over another and may carefully select (or create) a problem that illustrates the superiority of their personal preference. I call this process: creating a special bolt to fit a wrench you already have. Perhaps the best example of this is the High Performance Computing Center at nearby ORNL. They create problems to solve on absurdly expensive servers rather than showing how to cost-effectively solve important real-world problems (i.e., the tail wagging the dog). You can effectively use implicit or explicit methods to solve differential equations. In any case and whatever method you choose to use, don't presume that it will always work properly. Be careful and check the solution. Larger time steps with an implicit method may take equally as long (and perhaps longer) than smaller steps with an explicit method, yet arrive at the same solution so there is no simple answer to your question. Some clever methods have been developed to handle challenges that arise when solving such problems. One of the interesting approaches that's worth reading up on is the MacCormack scheme (alternating spatial differences to damp out unwanted oscillations that are a numerical artifact and not accurately representing the physics). You should also read up on Crank-Nicolson.
It is well known that implicit methods are more stable than the explicit ones. This point is very important specially for choosing large time steps.
This article on Implicit Runge-Kutta methods is a good read... https://www.epfl.ch/labs/anchp/wp-content/uploads/2018/05/part2-1.pdf I will pull together an example code illustrating the methods with a high resolution timer to compare computational effort for each method.
The first example in the article above is a good one. I solved it with explicit RK4 using 100, 125, 150, 175, and 200 steps. It's a good example of a stiff system because it goes from being unstable at 100 steps to stable and smooth at 200 steps. I have attached a spreadsheet with the results. With this simple example it's not worth timing, as it would be for a more complex example.
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