Differentiating results in us knowing how large y-output changes compared to qan amount of x-input in the function f(x)=y.
Integrating is, uh, just “the summation” of all y-outputs as a result of x-inputs.
Now, what does a DE tell me? And what does solving it do?
In: Mathematics
Let’s imagine that you have a bathtube with a leak in it. You want to take a bath without all of the water running out, so you are going to run the faucet to add more water into the tube to offset the leak. However, the amount of water in the tub determines how faster water is going to leak out through the leak. This kind of problem is know as a “related rates” problem and is a fairly simple example of a differential equation. What we want to solve for is the rate of water we need to add, but that is dependant the rate of change of the leak, which is depend on the current water level. So:
y’ = a*x’ + b*x
Where a and b are some constants. After solving the DE, we will get an equation for y’ that is dependant on just x.
This is a fairly simple example, but what DE’s tend to be really useful is solving motion control problems. The one variable you have control over is the acceleration applied to the motion and you want to keep the system as a specific velocity or distance. Vehicle control is the obvious application, but the same math works for pretty much any physical system. The circuitry keeping your oven at a constant temperature is doing the same kind of operations.
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