Science:Math Exam Resources/Courses/MATH215/December 2013/Question 07 (c)
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) •
Question 07 (c) 

Consider the following initial value problem for a firstorder autonomous ODE: (c) Again if , use Euler’s method with the larger step size to approximate the solution at time . Predict what will happen if you continue to use this scheme (with ) to approximate for large . Compare this with your result from part (a), and explain any discrepancies. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

What sort of behaviour does Euler's method with h=2 predict about for large t? What should happen analytically (based on your analysis in part (a))? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Implementing the same notation as in part (b), we have We observe that and that so that
Given the information we know from part (a) that solution curves passing through a point approach we anticipate that if these approximations were to carry on for a long time, the approximations would (likely) approach . This disagrees with the prediction in part (a) in that if , the exact solution curve should approach as . The step size of is too large a step to take to accurately resolve the exact solution of the initial value problem. 