Science:Math Exam Resources/Courses/MATH215/December 2013/Question 07 (c)/Solution 1

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.