Solving the differential equations involving SHM

[anirocks11]

What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?

 

[Dale]

It is easy to check if a solution works once you make a guess. So generally you just have experience with non damped systems, and you guess and check until you get the form that works.

 

[sophiecentaur]

guessing certain solutions

That is a real problem when dealing with 'the next step' in Maths, for all students. "Why did they choose to do it that way?" we ask. Finding the way to the next step is always based on past experience so a student can't expect to find it to be very obvious.
In the SHM case, 'they' looked amongst the simplest function that would fit the rules - i.e. repeating and continuous etc etc. A single frequency sine wave is the first thing that would come to mind for SHM and it happens to be the right solution. When trying to solve the equations for Waves, there are many more possibilities (square waves, sawtooth etc) and they also work but we tend to start with a sine wave again - but we have to remember it's not the only one in that case.