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 $t\to \mathrm{\infty }$. The step size of $h=2$ is too large a step to take to accurately resolve the exact solution of the initial value problem.