>>10622254Whoa, I don't think i have the time to write a wall of text that would be required to pull you up from your plane of confusion, but I'll try my best.
Let's use a physical system so we can picture everything in our minds. Take a spring or slinky, suspend it tied to the ceiling, carefully let go. What happens? Nothing moves. The slinky is pulled down by gravity, and it creates an opposing force through Hooke's law.
That's the limit of the general solution. The initial condition would be the point and velocity of the free end of the slinky at t=0. If you pull it down and let go, it will oscillate for a bit, until the friction eventually dampens the motion, and you'll end up with the same image as before.
So, the initial condition is a single set of properties, at one point in time or space.
A particular solution is the result of something different, because it is not only a single point in time or space. Again, using the slinky example: If you suspend the thing and then do nothing, that corresponds to a force diagram with a "= 0" on the other side, and, as previously mentioned, gravity and Hooke's spring force cancelling each other out on the other side of the equation.
Now imagine you don't let go, but maybe wiggle the free end around in a sinusoidal motion, maybe. Or you constantly hold it at a fixed point, without letting go (constant, non-zero input).
I hope you can put it together now:
- homogeneous solution: what the system naturally does
- particular solution: what the system does with an input you provide (Just accept that you can add those to get the complete solution, for now. Or, you can take a look at the proof yourself..)
- initial condition: from which point in phase-space does the solution you're interested in start?
There, coffee cup's empty, I sincerely hope that helped. Good Luck.
