Euler - Eulers Method Up to this point practically every...

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Euler’s Method Up to this point practically every differential equation that we’ve been presented with could be solved. The problem with this is that these are the exceptions rather than the rule. The vast majority of first order differential equations can’t be solved. In order to teach you something about solving first order differential equations we’ve had to restrict ourselves down to the fairly restrictive cases of linear, separable, or exact differential equations or differential equations that could be solve with a set of very specific substitutions. Most first order differential equations however fall into none of these categories. In fact even those that are separable or exact cannot always be solved for an explicit solution. Without explicit solutions to these it would be hard to get any information about the solution. So what do we do when faced with a differential equation that we can’t solve? The answer depends on what you are looking for. If you are only looking for long term behavior of a solution you can always sketch a direction field. This can be done without too much difficulty for some fairly complex differential equations that we can’t solve to get exact solutions. The problem with this approach is that it’s only really good for getting general trends in solutions and for long term behavior of solutions. There are times when we will need something more. For instance, maybe we need to determine how a specific solution behaves, including some values that the solution will take. There are also a fairly large set of differential equations that are not easy to sketch good direction fields for. In these cases we resort to numerical methods that will allow us to approximate solutions to differential equations. There are many different methods that can be used to approximate solutions to a differential equation and in fact whole classes can be taught just dealing with the various methods. We are going to look at one of the oldest and easiest to use here. This method was originally devised by Euler and is called, oddly enough, Euler’s Method. Let’s start with a general first order IVP (1 ) where f(t,y) is a known function and the values in the initial condition are also known numbers. From the second theorem in the Intervals of Validity section we know that if f and f y are continuous function then there is a unique solution to the IVP in some interval surrounding
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. So, let’s assume that everything is nice and continuous so that we know that a solution will in fact exist. We want to approximate the solution to (1) near . We’ll start with the two pieces of information that we do know about the solution. First, we know the value of the solution at from the initial condition. Second, we also know the value of the derivative at . We can get this by plugging the initial condition into f(t,y) into the differential equation itself. So, the derivative at this point is.
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