We may not be able to find a formula for the solution y = φ(x), but if f(x,y) is a continuous function, we do know that such a solution exists. Say we have a generic initial value problem of the form dy⁄ dx = f(x,y) with y(x 0) = y 0. It can also be used to get an estimated value for that solution at a given value for x. Euler's Method is also called the tangent line method, and in essence it is an algorithmic way of plotting an approximate solution to an initial value problem through the direction field. Euler's Formula: A Numerical MethodĮuler's Method is one of the simplest and oldest numerical methods for approximating solutions to differential equations that cannot be solved with a nice formula. It is the most easily-understood example of a numerical approach to solving differential equations. (Numerical analysis is, after all, an entire branch of mathematics!) We will explore a couple of numerical methods, beginning with a relatively simple method called Euler's Method. In this lab we shall try to give you the basic idea behind answers to both Q1 and Q2, although going there from our sketch here is a pretty big leap. How does drawode actually compute solution curves? After all, it is dangerous to run programs without having a solid idea how they work if you do, eventually you will make a serious mistake. Thinking about direction fields and their limitations leads us to two questions. By changing the axes in slopefield, we might be able to zoom in and improve this estimate, but that can be an awkward way to obtain a highly accurate answer. From the direction field, we can guess that it is somewhere around 4.3. For example, we might want to find out the value of y in our solution above when x = 10. Even with two- or three-dimensional systems, visual techniques might fail us when we want to get very precise information. Indeed, most technology problems involve high-dimensional systems of ODEs, so that it is impossible to visualize their solutions. Unfortunately, such graphical techniques are limited to three dimensions at best. Pictures like these are very informative for understanding the behavior of the solutions to even very complicated differential equations. What does that suggest to us about this solution curve?
When you run these commands, you will get a warning: MATLAB says it is "unable to meet integration tolerances without reducing the step size." In this case, the curve becomes so steep on the left side that the computer cannot accurately compute more points to plot in a reasonable time.
We've already seen in Assignment 2 how to use slopefield and drawode to get a picture of our solution. One way we can get an idea of what the solutions to this (or any) differential equation look like is with a direction field. Thus we will need to use the numerical methods mentioned above. We have no symbolic methods for solving this equation. Convince yourself of this with a pencil and paper. The differential equation in (1) is not linear (because of the y 3), nor is it separable or exact. Consider the following initial value problem: (1)