Use qualitative methods, studied in chapter 4, to sketch the solution to the initial value problem:

dy=y2, y(0)=1 dt 4

Your answer should include:

steady-state solutions.

The graph of y vs. y with stability arrows is drawn.

The stability of steady-state(s) labeled.

The solution curve, y(t) vs t.

If applicable, clear indications of any asymptotic behavior, concavity, and points of inflection.

(b) Use separation of variables to solve the initial value problem in part (a).

Observe that the solution has a vertical asymptote at t = 4. Draw a vertical dashed line at t = 4 to highlight the vertical asymptote. This phenomenon is called blow-up because the solution becomes infinite in a finite time. Note that the blow-up cannot be predicted by qualitative methods (done in part (a)).

(c) Use the workbook ode.xls, found at https://math.sci.ccny.cuny.edu/course/math-20900/

SELECT math209files > Analysis folder AND SELECT ODE.XLS to solve the initial value problem in part (a) using the Modified Euler's Method. In the ODE Editor pop-up, set Final time = 4 and # of subintervals = 50.