We’re taking very basic differential equations in school and I know how to solve them (more or less) but I think I’d be better when I u derstand what this is doing. Like we use differentiation to find the slope of a curve integration to find the curve equation from a given slope of a point or find the area under the curve
When I solve a differential equation. What is it that I do?
In: Mathematics
Well, I like to remember that Newton invented calculus to explain things in motion. So a practical example often helps.
Like we know distance as here to there.
And speed is d = r/t
And Acceleration is v = r/t/t
Those are differential moments. If we know your position, given by some function, then your speed can be derived from distance from the initial point as a function of time. and your rate of speed over time tells use your positional Acceleration.
So what the differential calculus is helping us to understand is that if your position changes we can figure it out where you are by the rate of change over time. And even more accurately if we know the rate of change at a specific time (or as your rate of velocity changes over time)
And, in a concrete example. If you throw a ball, and it’s arc in space is described by some function. We can calculate its distance, velocity and speed at different points in time, using differential calculations (the ball goes up, but is acted on by wind and gravity) so it’s not a straight line – but a function that we can manipulate
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