1. CHAPTER 2 2.4 Continuity Euler’s Method Euler’s idea was to improve on linear approximation by proceeding only a short distance along the tangent line at ( x, y’(x)) and then making a midcourse correction by changing the direction as indicated by the direction field. In general, Euler’s method says to start at the point given by the initial value and proceed in the direction indicated by the direction field. Stop after a short time, look at the slope at the new location, and proceed in that direction. Keep stopping and changing the direction according to the direction field.

2. Euler’s Method Numerical method of finding approximate solutions of differential equations. y x

3. Euler’s Method y x Consider the initial value problem: y’ = F(x,y) y(x 0 )= y 0

4. Euler’s Method y x Consider the initial value problem: y’ = F(x,y) y(x 0 )= y 0 (x 0,y 0 ) y(x)

5. Euler’s Method y x Consider the initial value problem: y’ = F(x,y) y(x 0 )= y 0 (x 0,y 0 ) y0y0 x0x0 y(x)

6. Euler’s Method y x Consider the initial value problem: y’ = F(x,y) y(x 0 )= y 0 Slope =F (x 0,y 0 ) (x 0,y 0 ) y0y0 x0x0 (x 1,y 1 ) y(x)

7. Euler’s Method y x Consider the initial value problem: y’ = F(x,y) y(x 0 )= y 0 (x 0, y 0 ) y0y0 x0x0 h x1x1 (x 1, y 1 ) Slope =F (x 0,y 0 ) hF (x 0,y 0 ) y(x) x x 1 = x 0 + h

8. Euler’s Method y x Consider the initial value problem: y’ = F(x,y) y(x 0 )= y 0 (x 0, y 0 ) y0y0 x0x0 h x1x1 (x 1, y 1 ) Slope =F (x 0,y 0 ) hF (x 0,y 0 ) y(x) x x 1 = x 0 + h y 1 = y 0 + hF (x 0,y 0 )

9. y x (x 0, y 0 ) y0y0 x0x0 h x1x1 (x 1, y 1 ) Slope =F (x 1,y 1 ) hF (x 0,y 0 ) y(x) x x 1 = x 0 + h y 1 = y 0 + hF (x 0,y 0 ) h (x 2, y 2 ) x 2 = x 1 + h

10. y x (x 0, y 0 ) y0y0 x0x0 h x1x1 (x 1, y 1 ) Slope =F (x 1,y 1 ) hF (x 0,y 0 ) y(x) x x 1 = x 0 + h y 1 = y 0 + hF (x 0,y 0 ) h (x 2, y 2 ) x 2 = x 1 + h hF (x 1,y 1 ) y 3 = y 2 + hF (x 2,y 2 ) x 3 = x 2 + h y 2 = y 1 + hF (x 1,y 1 )

11. y x (x 0, y 0 ) y0y0 x0x0 h x1x1 (x 1, y 1 ) Slope =F (x 1,y 1 ) hF (x 0,y 0 ) y(x) x x 1 = x 0 + h y 1 = y 0 + hF (x 0,y 0 ) h (x 2, y 2 ) x 2 = x 1 + h hF (x 1,y 1 ) y 3 = y 2 + hF (x 2,y 2 ) x 3 = x 2 + h y 2 = y 1 + hF (x 1,y 1 ) x n = x n-1 + h y n = y n-1 +hF(x n-1,y n-1 ) x n = x n-1 + h y n = y n-1 +hF(x n-1,y n-1 )

12. CHAPTER 2 2.4 Continuity Example: Use Euler’s method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem y’ = x + y 2, y(0) = 0.