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.

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.

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

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

Example can be represented by tangent line segments

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

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

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.

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

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

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

Try This Which choice represents the slope field for A B

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

Applet link

Applet link GSP slopefield link