MA/CS 371 Final Exam, April 25, 1996

Directions: This examination is closed book and notes. You may have only one 8.5 (inches) by 11 (inches) sheet of paper containing any review material (front and back) in addition to the examination sheet and solution sheets. To receive credit, you must show all your work. If you find it necessary to make assumptions, please explicitly state them. You have 75 minutes to complete this examination. Show all your work on the solution sheets provided (you may keep the exam sheet) and make sure all the solution sheets remain stapled together before handing them in.

1. [20 Pts] Calculate an approximate value for x(0.5) where the function x(t) satisfies the ordinary differential equation

                 -         2     3
                |  x'   = t  +  x
                |  x(0) = 0 
                 -
       

   (a) using one step of the Taylor Series method of order 2,

   (b) using one step of the Runge-Kutta method of order 2 (i.e., RK2).

2. [20 Pts] Complete the following Matlab function euler.m which could be used to solve the following initial-value problem (IVP) for x(b).

                 -         
                |  x'   = f(t,x(t))
                |  x(a) = s 
                 -
       

   using n steps of size h=(b-a)/n. You may assume the existence of an external Matlab function func.m which is invoked via func(t,x) to evaluate the function f(t,x(t)) in the IVP.

   function [e] = euler(n,a,b,s) h = (b-a)/n;

   Turn page over ...

3. [20 Pts] Write an equivalent autonomous system of first-order differential equations for the following coupled system of first-order ordinary differential equations.

                 -          3    y
                |  x'   =  x  + e  + t
                |                              2
                |  y'   =  log x  - cos y + t x
                |                             
                |  x(0) = 1, y(0) = 2
                 -
       

4. [20 Pts] Rewrite the following third-order ordinary differential equation as an autonomous system of first-order ordinary differential equations having the form X'=F(X), X(1)=S, where X, X', F, and S are all vectors.

                 -         
                |  x''' = 2 x'' + cos(tx) - x'/t
                |  x(1) = 1, x'(1) = 0, x''(1) = 2 
                 -
       

5. [20 Pts] Sketch the approximations generated by the Euler and RK2 methods to the solution curve x(t) shown on the last page of the answer sheets for t = a+ih, where a=1.0, h=1.0, and i=0,1,2. That is, you must illustrate 3 approximate values for the the solution x(t) by each method. Label each approximation clearly.

Total points possible on this exam : 100 Points