Problem 2: Consider following initial value problem: dy =k(a - y)(b - y) y(0) = 0...

Transcribed Image Text:

Problem 2: Consider following initial value problem: dy =k(a - y)(b - y) y(0) = 0 dt where k = 0.01, a = 70 and b = 50 and the exact solution is y(t) = 350 [1-exp(-0.21)] /[7-5 exp(-0.2t)] a) Verify that this is the solution, by substituting into the original differential equation. b) Use the method of separation of variables, to derive the solution. c) Write a MATLAB code to numerically solve the initial value problem over the interval [0, 20] using Euler's method with a fixed sample period, t = 0.5. Calculate the error curve for the following sample periods t = 0.5, 0.2 and 0.1. d) Repeat part a) using Heun's method and the Runge- Kutta method. Problem 2: Consider following initial value problem: dy =k(a - y)(b - y) y(0) = 0 dt where k = 0.01, a = 70 and b = 50 and the exact solution is y(t) = 350 [1-exp(-0.21)] /[7-5 exp(-0.2t)] a) Verify that this is the solution, by substituting into the original differential equation. b) Use the method of separation of variables, to derive the solution. c) Write a MATLAB code to numerically solve the initial value problem over the interval [0, 20] using Euler's method with a fixed sample period, t = 0.5. Calculate the error curve for the following sample periods t = 0.5, 0.2 and 0.1. d) Repeat part a) using Heun's method and the Runge- Kutta method.