1. Problem: we can’t solve the differential equation!!!
Suppose we have the initial value problem:
Problem: we can’t solve the differential equation!!!
It’s not separable
and the initial condition:
Find a formula for y at any time t.

2. Important Idea The slope
(rate of change) at any point (x,y) on the solution curve is the x coordinate of the point minus the y coordinate.

3. Solution Curve
Example
(0,1)
Rate of change at (0,1)=x-y=-1

4. Solution Curve
Example
(2,1)
Rate of change at (2,1)=x-y=1

5. Example
can be represented by tangent line segments

6. Definition
All such segments represent the slope field or direction field for

7. Example Using the slope field, sketch the solution curve through (0,1)
Hint: start at (0,1). Sketch right then left,

8. Try This
Using the slope field, sketch the solution curve through (1,0)
(1,0) is the initial condition. Estimate the solution to the initial value problem at x=3.

9. Example
For
Sketch the tangent line segments (slope field) at each integer coordinate

10. Important Idea
Sketching slope fields can be tedious. It is best done with a graphing program.

11. Try This This is the slope field for
Confirm that the solution curve is
Hint: Solve the D.E.

12. Try This
Which choice represents the slope field for A B

13. Try This
This slope field is for which differential equation? A B C

14. Applet link

15. Applet link
GSP slopefield link