Eulers Method CHAPTER 2 Eulers idea was to

Euler’s Method CHAPTER 2 Euler’s idea was to Continuity improve on linear 2. 4 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.

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

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

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

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

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

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

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

Slope =F (x 1, y 1 ) (x 2, y 2 ) x 1= x 0+ h y 1= y 0+ h. F (x 0, y 0) x 2= x 1+ h y y(x) (x 0, y 0 ) h (x 1, y 1 ) h. F (x 0, y 0 ) h y 0 x 1 x

Slope =F (x 1, y 1 ) (x 2, y 2 ) y(x) h. F (x 1, y 1 ) y (x 0, y 0 ) x 1= x 0+ h y 1= y 0+ h. F (x 0, y 0) x 2= x 1+ h y 2= y 1+ h. F (x 1, y 1) x 3= x 2+ h y 3= y 2+ h. F (x 2, y 2) h (x 1, y 1 ) h. F (x 0, y 0 ) h y 0 x 1 x

Slope =F (x 1, y 1 ) y (x 2, y 2 ) y(x) h. F (x 1, y 1 ) (x 0, y 0 ) h (x 1, y 1 ) h. F (x 0, y 0 ) x 1= x 0+ h y 1= y 0+ h. F (x 0, y 0) x 2= x 1+ h y 2= y 1+ h. F (x 1, y 1) x 3= x 2+ h y 3= y 2+ h. F (x 2, y 2) xn= xn-1+ h yn= yn-1+h. F(xn-1, yn-1) h y 0 x 1 x

Example: Use Euler’s method with step size 0. 2 to CHAPTER estimate y(1), where y(x) is the 2 solution of the initial-value 2. 4 problem y’ = x + y 2 , y(0) = 0. Continuity