The population will grow faster and faster.
But that is only true at a specific time , and doesn't include that the population is constantly increasing. So it is better to say the rate of change at any instant is the growth rate times the population at that instant:. And that is a Differential Equation , because it has a function N t and its derivative. And how powerful mathematics is!
That short equation says "the rate of change of the population over time equals the growth rate times the population". Differential Equations can describe how populations change, how heat moves, how springs vibrate, how radioactive material decays and much more. They are a very natural way to describe many things in the universe. On its own , a Differential Equation is a wonderful way to express something, but is hard to use.
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. Money earns interest. The interest can be calculated at fixed times, such as yearly, monthly, etc. But when it is compounded continuously then at any time the interest gets added in proportion to the current value of the loan or investment.
Using t for time, r for the interest rate and V for the current value of the loan:. And here is a cool thing: it is the same as the equation we got with the Rabbits! It just has different letters. So mathematics shows us these two things behave the same.
But don't worry, it can be solved using a special method called Separation of Variables and results in:. So Differential Equations are great at describing things, but need to be solved to be useful. In Physics, Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement.
An example of this is given by a mass on a spring. The weight is pulled down by gravity, and we know from Newton's Second Law that force equals mass times acceleration:. And acceleration is the second derivative of position with respect to time, so:.
- Differential equation!
- MA Differential Equations?
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- Differential equations;
- Introduction to Differential Equations;
It has a function x t , and it's second derivative d 2 x dt 2. 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 :. Creating a differential equation is the first major step. But we also need to solve it to discover how, for example, the spring bounces up and down over time.
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Over the years wise people have worked out special methods to solve some types of Differential Equations. It is like travel: different kinds of transport have solved how to get to certain places. Is it near, so we can just walk? An ordinary differential equation ODE is an equation that involves some ordinary derivatives as opposed to partial derivatives of a function.
Often, our goal is to solve an ODE, i. If you know what the derivative of a function is, how can you find the function itself?
Differential equations introduction
You need to find the antiderivative, i. Except we forgot one important point: there is always an arbitrary constant that we cannot determine if we only know the derivative. In general, solving an ODE is more complicated than simple integration. Even so, the basic principle is always integration, as we need to go from derivative to function.
Usually, the difficult part is determining what integration we need to do. Let's start simpler, though. What is the simplest possible ODE? We can ask some simple questions. If not, what else do you need to specify?
Differential Equations - Introduction
Let's make things a little more complicated. The solution is just the antiderivative, or the integral. Let's do the integral slightly differently this time. We can write the solution in different ways.