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
Differential equations are different from differentiation and integration, but they’re related.
Imagine you have a boiling pot of water that you lay out on the table. You’re curious about how fast it will cool down. Your natural intuition tells you that it’s cooling down faster when the water is still really hot, but cools slower as the water gets closer to room temperature.
You would be right this is called Newton’s law of cooling. It states that the rate of cooling of an object is proportional to how much difference there is between the object and the environment. So let’s write out that equation. T’ will be the rate of change of temperature, A will be the temperature of the room, and T will be temperature of the pot of water.
T’=(A-T)
Seems supper easy right, but take a closer look. That equation has the speed temperature is changing, and also temperature. That’s a differential equation. When your rate of change is determined by the thing that’s changing.
Now what would be really useful is if we could somehow figure out a formula based on time from that. You can do it, and that’s what you’ll learn in a differential equations class.
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