Slope fields

[abacus]

I'm still having trouble with figuring out slope fields.

If I have a function dy/dt= f(t,y), how would I go about graphing the slope field for my solutions without the use of a calculator.

An example would be how would I graph the slope field of dy/dt = t^2-t and dy/dt = y^2? Thanks.

[HallsofIvy]

Choose a number of (t,y) points on your graph. Calculate f(t,y) for each of those. Draw a short line through (t,y) with slope f(t,y).

For example, at (1,1) with dy/dt= t2- t, dy/dt= 0 so we draw a horizontal line- in fact, since y is not explicitely in that formula, it is obvious that dy/dt= 0 for all points on the vertical line t=1. At each point on that line, draw a short horizontal line. At t= 0.5, t2= .25-.5= -.25 so at every point on the vertical line t= 0.5, draw a short line with slope -.25. Try to connect those to the lines at t= 1.

For dy/dt= y2, since t does not appear explicitely, you can calculate y2 for each y and draw short lines with that slope all along the horizontal line at y.