Differential equations what is the purpose

But that is only true at a specific time , and doesn't include that the population is constantly increasing. The bigger the population, the more new rabbits we get! 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!

Calc 2 is just brutal hard work, trig identities and decomposition integrals that are really just a rite of passage. Of all the calculuses, calc 2 is the hardest. Discrete mathematics is actually really straightforward. This hypothesis is thought to be the most important unsolved problems in mathematics, let alone in Calculus. The process of finding the derivative of any given function is known as differentiation.

Begin typing your search term above and press enter to search. Press ESC to cancel. Ben Davis April 15, What is the purpose of differential equations? Should I take differential equations? Of all the calculuses, calc 2 is the hardest.

Discrete mathematics is actually really straightforward. This hypothesis is thought to be the most important unsolved problems in mathematics, let alone in Calculus. The process of finding the derivative of any given function is known as differentiation. Begin typing your search term above and press enter to search.

Press ESC to cancel. Ben Davis April 15, What is the purpose of differential equations? Should I take differential equations? Is Calc 3 easier than differential equations? This is one of the first differential equations that you will learn how to solve and you will be able to verify this shortly for yourself. The actual solution to a differential equation is the specific solution that not only satisfies the differential equation, but also satisfies the given initial condition s.

This is actually easier to do than it might at first appear. From the previous example we already know well that is provided you believe our solution to this example… that all solutions to the differential equation are of the form. To find this all we need do is use our initial condition as follows.

From this last example we can see that once we have the general solution to a differential equation finding the actual solution is nothing more than applying the initial condition s and solving for the constant s that are in the general solution. At this point we will ask that you trust us that this is in fact a solution to the differential equation.

You will learn how to get this solution in a later section. There are two functions here and we only want one and in fact only one will be correct! We can determine the correct function by reapplying the initial condition.

Only one of them will satisfy the initial condition. The actual explicit solution is then.