Ms. Battaglia AP Calculus

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

2. 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

3. 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 (x0,y0). Starting point: the graph of the solution passes through the point (x0,y0) and has the slope F(x0,y0). 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 (x1,y1) where x1 = x0 + h and y1 = y0 + hF(x0,y0) If you think of (x1,y1) as a new starting point, you can repeat the process to obtain a second point (x2,y2)

4. Euler’s Method x1 = x0 + h y1 = y0 + hF(x0,y0) x2= x1+ h y2= y1+ hF(x1,y1) . . . . . . xn= xn-1+ h yn= yn-1+ hF(xn-1,yn-1)

5. 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.

6. 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.

7. Approximating a Solution Using Euler’s Method Use Euler’s Method to approximate the particular solution of the differential equation y ‘ = 3x – 2y passing through the point (0,3). Use a step of h=0.05 and n=10.

8. 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)