2. Solve the following pair of ODEs over the interval from t = 0 to 0.4 using a step size of 0.1.The initial conditions are y(0) = 2 and z(0) = 4. Obtain your solution witha. Euler's method andb. the second-order RK method.- 2y + 5etdt12²dtII1

There are may formula for Runge-kutta of order two, i use the modified Euler form

Answered: 2. Solve the following pair of ODEs…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find the solution for the following ODE; y2 - y' = 0 with the initial condition y(6) = 3 What is the value of the constant C ?
Apply the improved Euler method to approximate the solution on the interval [0, 0.5] with step size h= 0.1. Construct a table showing values of the approximate solution and the actual solution at the points x 0.1, 0.2, 0.3, 0.4, 0.5. y =y-x-3, y(0) = 1; y(x) = 4 +x-3 e* Complete the table below. (Round to four decimal places as needed.) 0.1 0.2 0.3 0.4 0.5 Actual, y (Xn) Improved Euler, yn
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A particular region has a rabbit population of 1600. Two foxes are introduced to control the population of rabbits. Following this, the number of rabbits decreases according to the formula R(t) = 1700 – Aekt. - where A and k are constants, and R(t) is the number of rabbits in the region t years after the introduction of the foxes. (a) Given that the population of rabbits drops by one quarter after 5 years, find the values of A and k. (b) Following this model, how long will it take for the rabbits to become extinct? Give your answer to two decimal places. (c) Let F(t) be the number of foxes in the region t years after their introduction. If dF = 0.7F(t), dt find the time at which the rate of decrease of the rabbit population is equal to the rate of increase of the fox population, correct to two decimal places. dR dF Hint. Note that and represent the rates of change of the rabbit and fox dt dt populations respectively. (d) Identify any problems with this model.
Part 2. Consider an ODE of the form x²y" + axy' + by = 0 with given constants a and b and unknown solution y(x). Assuming that y(x) follows the form y = xm, perform the following tasks: 2A. Solve for y'(x) and y" (x) and transform the given ODE in terms of x, m, a, and b. 2B. Determine the characteristic equation of the ODE. For each solution listed below, determine the corresponding roots of the characteristic equation and derive the respective Cauchy-Euler ODE: -3 2C. y = x ³ +1 2D. y = (1 + Inx)x` 2E. y = x[cos(2lnx) + sin(2lnx)]
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2. Solve the following pair of ODEs over the interval from t = 0 to 0.4 using a step size of 0.1. The initial conditions are y(0) = 2 and z(0) = 4. Obtain your solution with a. Euler's method and b. the second-order RK method. - 2y + 5et dt 12² dt II 1