6 1 Eulers Method Objective Use Eulers Method

6 -1 Euler’s Method Objective: Use Euler’s Method to approximate solutions of differential equations. Ms. Battaglia AP Calculus

Estimate y(4) with a step size h=1, where y(x) is the solution to the initial value problem: y’ – y = 0 ; y(0) = 1

Euler’s Method Euler’s method is a numerical approach to approximating the particular solution of the differential equation y ’ = F(x, y) that passes through the point (x 0, y 0). Starting point: the graph of the solution passes through the point (x 0, y 0) and has the slope F(x 0, y 0). Next, proceed in the direction indicated b the slope. Using a small step h, move along the tangent line until you arive at the point (x 1, y 1) where x 1 = x 0 + h and y 1 = y 0 + h. F(x 0, y 0) If you think of (x 1, y 1) as a new starting point, you can repeat the process to obtain a second point (x 2, y 2)

Euler’s Method x 1 = x 0 + h x 2 = x 1 + h. . . xn = xn-1 + h y 1 = y 0 + h. F(x 0, y 0) y 2 = y 1 + h. F(x 1, y 1). . . yn = yn-1 + h. F(xn-1, yn-1)

Approximating a Solution Using Euler’s Method Use Euler’s Method to approximate the particular solution of the differential equation y‘ = x – y passing through the point (0, 1). Use a step of h=0. 1 and n=10.

Approximating a Solution Using Euler’s Method Use Euler’s Method to approximate the particular solution of the differential equation y‘ = x + y passing through the point (0, 2). Use a step of h=0. 1 and n=10.

Approximating a Solution Using Euler’s Method Use Euler’s Method to approximate the particular solution of the differential equation y ‘ = 3 x – 2 y passing through the point (0, 3). Use a step of h=0. 05 and n=10.

Classwork/Homework � AB: Page 413 #73, 74, 77, 78 (use n = 5 for all 4 problems), 79, 89 -92 � BC: Page 413 #73 -78, 89 -92, and Worksheet(matching)