Question

5. [25 pointsl Given the initial value problem with the initial conditions y(0) 2 and y'(0)10, (a) Solve analytically to obtain the exact solution y(x) (b) Solve numerically using the forward Euler, backward Euler, and fourth-order Runge Kutta methods. Please implement all three methods yourselves do not use any built- in integrators (i.e., ode45)). Integrate over 0 3 r < 4, and compare the methods with the exact solution. (For example, using a table and/or plots.) Vary the step size used for each method over the values δχ = 0.1, 0.2. and 0.3: for which values are each of the methods stable or unstable? (c) At what value of step size (to the hundredth) does the forward Euler method first (d) At what value of step size (to the hundredth) does the 4th-order Runge-Kutta method (e) Does the backward Euler method become unstable for large step size values? How show signs of instability? When does it become "fullv unstable become unstable? does the error of the method (based on comparison with the exact solution) increase with growing step size? (f) Now compare the performance and accuracy of the three methods using the step size Ar 0.1. Which method would you recommend using?

The Program for the code should be matlab

Answer 1

ven (ienssal 3olukon T ind CF, here j(x) = lox+11 D HID+IO 10久 2. D+IID+to

PI- Go lo久 2 (D+ 0410) LA

1 I O l O -1010 ーズ

2 Solubon