Science:Math Exam Resources/Courses/MATH215/December 2013/Question 07 (c)

MATH215 December 2013

  •  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)  •  

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Question 07 (c)

Consider the following initial value problem for a first-order autonomous ODE: ${\frac {dy}{dt}}=-y(y-1)^{2},\quad y(0)=y_{0}$ (c) Again if $y_{0}=3/2$, use Euler’s method with the larger step size $h=2$ to approximate the solution $y(2)$ at time $t=2$. Predict what will happen if you continue to use this scheme (with $h=2$) to approximate $y(t)$ for large $t$. 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 $y(t)$ 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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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 $y(2)\approx y_{1}=y_{0}+h[-y_{0}(y_{0}-1)^{2}]=3/2+2(-3/8)=3/4.$ We observe that $0<y_{1}<1$ and that ${\frac {dy}{dt}}|_{y=3/4}<0$ so that $y_{2}=y_{1}+h{\frac {dy}{dt}}|_{y=3/4}<y_{1}$. Given the information we know from part (a) that solution curves passing through a point $y_{0}<1$ approach $y=0$ we anticipate that if these approximations were to carry on for a long time, the approximations would (likely) approach $y=0$. This disagrees with the prediction in part (a) in that if $y(0)=3/2$, the exact solution curve should approach $y=1$ as $t\rightarrow \infty$. The step size of $h=2$ is too large a step to take to accurately resolve the exact solution of the initial value problem.