Solving Numerical Differential Equations using Euler's Method - Prof. J. M. Mahaffy, Assignments of Mathematics

Download Solving Numerical Differential Equations using Euler's Method - Prof. J. M. Mahaffy and more Assignments Mathematics in PDF only on Docsity! Fall 2000 Problems Numerical Differential Equations 1. Consider the following initial value problems: a. dy dt = 0.3y, y(0) = 20. b. dy dt = 10− 0.3y, y(0) = 10. Solve each of these initial value problems, then use Euler’s method to approximate the solution using a stepsize of h = 0.2 for t ∈ [0, 1]. Find the approximate solution for y(1), then compute the percent error between the actual solution and the approximate solution using Euler’s method. 2. A population of animals that includes emigration satisfies the differential equation P ′ = kP −m, P (0) = 100, where k = 0.1 and m = 2. a. Solve this differential equation and find P (1). b. Use Euler’s method with h = 0.2 to approximate the solution at t = 1. Find the percent error between the actual solution and this approximate solution at t = 1. 3. The temperature of an object is initially 50◦C. If it is in a room where the temperature, Te(t), is slowly decreasing with Te(t) = 20− t, then using Newton’s Law of Cooling, the temperature of the object satisfies the differential equation T ′ = −k(T − (20− t)), where k = 0.2 hr−1. a. Verify that the solution to this initial value problem is given by T (t) = 25− t + 25e−0.2t and find the temperature at t = 2. b. Use Euler’s method with h = 0.5 to approximate the solution at t = 2. This means that you are to the differential equation above and take four Euler’s method steps. Find the percent error between the actual solution and this approximate solution at t = 2.