1. 1 6.1 Slope Fields and Euler's Method Objective: Solve differential equations graphically and numerically.

2. 2 Slope Fields: a graphical approach Solving some DEs can be difficult or even impossible  Take a graphical approach Consider y’= F ( x, y )  At each point (x, y) in the xy-plane where F is defined, the DE determines the slope y’= F ( x, y ) of the solution at that point.  If you draw a short line segment with slope F ( x, y ) at selected points (x, y) in the domain of F, these line segments form a slope field.  A slope field shows the general shape of all the solutions

3. 3 Sketch the slope field for the differential equation y’= x - y

4. 4

5. 5 Match the slope field to the DE

6. 6 Sketch the slope field for the differential equation y’= 2x + y Use your slope field to sketch the solution that passes through (1,1)

7. 7 Sketch the slope field for the differential equation y’= 2x + y Use your slope field to sketch the solution that passes through (1,1)

8. 8 Euler’s Method: a numerical approach Let y’ = f (x,y) be a differential equation with initial condition y = y o when x = x o Partition the x-axis to the right of x o into subintervals x o < x 1 < x 2 < … of length Euler’s Method approximates the values of the exact points using the known slope of the tangent line at ( x o, y o ) to estimate the solution y 1 at x 1

9. 9 Repeat using ( x 1, y 1 ) to estimate the solution y 2 at x 2, using ( x 2, y 2 ) to estimate the solution y 3 at x 3 and so on….

10. 10 Example

11. 11 The first ten approximations are shown in the table. You can plot these values to see a graph of the approximate solution.