![]() It is like travel: different kinds of transport have solved how to get to certain places. So we need to know what type of Differential Equation it is first. Over the years wise people have worked out special methods to solve some types of Differential Equations. But we also need to solve it to discover how, for example, the spring bounces up and down over time. Note: we haven't included "damping" (the slowing down of the bounces due to friction), which is a little more complicated, but you can play with it here (press play):Ĭreating a differential equation is the first major step. It has a function x(t), and it's second derivative d 2x dt 2 The spring pulls it back up based on how stretched it is ( k is the spring's stiffness, and x is how stretched it is): F = -kx The weight is pulled down by gravity, and we know from Newton's Second Law that force equals mass times acceleration:Īnd acceleration is the second derivative of position with respect to time, so: then it falls back down, up and down, again and again.then the spring's tension pulls it back up,.as the spring stretches its tension increases,.the weight gets pulled down due to gravity,.So we try to solve them by turning the Differential Equation into a simpler equation without the differential bits, so we can do calculations, make graphs, predict the future, and so on. On its own, a Differential Equation is a wonderful way to express something, but is hard to use. They are a very natural way to describe many things in the universe. Find the average value of the function f(x) x 2 over the interval 0, 6 and find c such that f(c) equals the average value of the function over 0, 6. So it is better to say the rate of change (at any instant) is the growth rate times the population at that instant:Īnd that is a Differential Equation, because it has a function N(t) and its derivative.Īnd how powerful mathematics is! That short equation says "the rate of change of the population over time equals the growth rate times the population".ĭifferential Equations can describe how populations change, how heat moves, how springs vibrate, how radioactive material decays and much more. Figure 5.3.1: By the Mean Value Theorem, the continuous function f(x) takes on its average value at c at least once over a closed interval. When the population is 2000 we get 2000×0.01 = 20 new rabbits per week, etc. The bigger the population, the more new rabbits we get! When the population is 1000, the rate of change dN dt is then 1000×0.01 = 10 new rabbits per week.īut that is only true at a specific time, and doesn't include that the population is constantly increasing. Let us imagine the growth rate r is 0.01 new rabbits per week for every current rabbit. Think of dN dt as "how much the population changes as time changes, for any moment in time". Then those rabbits grow up and have babies too! The population will grow faster and faster. The more rabbits we have the more baby rabbits we get.
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