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Unformatted text preview: Math 216: Differential Equations Lab 3: Higher-Order Numerical Methods, Linearity and Superposition Goals In this lab we have two goals: we will first examine some new numerical methods to find approximations to the solutions to differential equation: that is, we will test-drive some new numerical solvers. We will implement an improved Eulers method, and will use Matlabs built in numerical solver which is an implementation of the Runge-Kutta method. Our second goal is to use these solvers to explore solutions to a linear and a nonlinear differential equation. This will allow us to see how superposition works (for linear equations) and doesnt (for nonlinear ones). Application: a pendulum The angular position of a simple pendulum can be described by differential equations. Con- sider the pendulum shown below. If measures the angular displacement of the pendulum bob from the vertical, then 00 ( t ) + g L sin( ( t )) = f ( t ) , (1) where g is the acceleration due to gravity, L is the length of the pendulum, and f ( t ) is any forcing imposed on the system. For this lab we will consider f ( t ) = 0. Note that this equation is conspicuously nonlinear ! As a result, we are unable to solve it in terms of elementary functions. Because of this, we often use its linearization, 00 ( t ) + g L ( t ) = f ( t ) , (2) which is valid for small angles and is obviously much easier (possible!) to solve. For this lab, we will take f ( t ) = 0, g = 9 . 81 m/s 2 , and L = 1 m (quite a big pendulum). 1 Prelab assignment Before arriving in the lab, answer the following questions. You will need your answers in lab to work the problems, and your recitation instructor may check that you have brought them. These problems are to be handed in as part of your lab report. First, consider the linearized equation modeling the pendulums motion, (2), with the values indicated: 00 ( t ) + 9 . 81 ( t ) = 0 . 1. Suppose that we pull the pendulum back an angle of 4 and then release it. Write the initial conditions that correspond to this physical situation, and then solve the linearized pendulum equation with these initial conditions. 2. Next, suppose that instead of pulling the pendulum back and releasing it, we give it a tap so that it has an initial velocity, say 2 m/s. Write out the initial conditions for this physical situation, and then solve the linearized pendulum equation with these initial conditions. 3. Now consider the combination of these initial conditions: we pull the bob of the pen- dulum back an angle of 4 and also push it outwards with a speed of 2 m/s. Write down the initial conditions in this case. Then note that you can write down a solution to this initial problem using your work in the preceding two questions. Why are you able to do this? What is the solution to this initial value problem?...
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